Newton's Principia : the mathematical principles of natural philosophy

MATH.-STAT.

SIM ISAAC MIBWf OM«

NEWTON'S PRINCIPIA.

THE

MATHEMATICAL PRINCIPLES

OF

NATURAL PHILOSOPHY,

BY SIR ISAAC NEWTON;

TRANSLATED INTO ENGLISH BY ANDREW MOTTE.

TO WHICH IS ADDKTV

NEWTON'S SYSTEM OF THE WORLD ;

With a Portrait taken from the Bust in the Royal Observatory at Greenwich.

FIRST AMERICAN EDITION, CAREFULLY REVISED AND CORRECTED,

WITH A LIFE OF THE AUTHOR, BY PI. W. CHITTENDEN, M. A., &e.

NEW-YORK

PUBLISHED BY DANIEL ADEE, 45 LIBERTY STREET.

p*-

Kntered according to Act of Congress, in the year 1846, by

DANIEL ADEE.
3!Ltht Clerk's Office ut'tiie Southern Oisli:ct Court of New-York.

TWuey * Lockwoof, Stom
16 Spruce St. N. Y.

DEDICATION.

TO THE

TEACHERS OF THE NORMAL SCHOOL

OF THE STATE OF NEW-YORK.

GENTLEMEN !

A stirring freshness in the air, and ruddy streaks upon the
horizon of the moral world betoken the grateful dawning of a new
ora. The days of a drivelling instruction are departing. With
us is the opening promise of a better time, wherein genuine man
hood doing its noblest work shall have adequate reward.
TEACHER is the highest and most responsible office man can fill.
Its dignity is, and will yet be held commensurate with its duty —
a duty boundless as man's intellectual capacity, and great as his
moral need — a duty from the performance of which shall emanate
an influence not limited to the now and the here, but which surely
will, as time flows into eternity and space into infinity, roll up, a
measureless curse or a measureless blessing, in inconceivable
swellings along the infinite curve. It is an office that should be
esteemed of even sacred import in this country. Ere long a hun
dred millions, extending from the Atlantic to the Pacific, from
Baffin's Bay to that of Panama, shall call themselves American
citizens. What a field for those two master-passions of the hu
man soul — the love of Rule, and the love of Gain ! How shall
our liberties continue to be preserved from the graspings of Am
bition and the corruptions of Gold ? Not by Bills of Rights

4 DEDICATION.

Constitutions, and Statute Books ; but alone by the rightly culti
vated hearts and heads of the PEOPLE. They must themselves
guard the Ark. It is yours to tit them for the consecrated
charge. Look well to it : for you appear clothed in the majesty
of great power ! It is yours to fashion, and to inform , to save,
and to perpetuate. You are the Educators of the People : you
are the prime Conservators of the public weal. Betray your
trust, and the sacred fires would go out, and the altars crumble
into dust : knowledge become lost in tradition, and Christian no
bleness a fable ! As you, therefore, are multiplied in number,
elevated in consideration, increased in means, and fulfill, well and
faithfully, all the requirements of true Teachers, so shall our fa
voured land lift up her head among the nations of the earth, and
call herself blessed.

In conclusion, Gentlemen, to you, as the conspicuous leaders
in the vast and honourable labour of Educational Helbrm, ana
Popular Teaching, the First American Edition of the PRINCIPIA ol
Newton — the greatest wrork of the greatest Teacher — is most
respectfully dedicated.

N. W. CHITTENDEN.

INTRODUCTION TO THE AMERICAN EDITION.

THAT the PRINCIPIA of Newton should have remained so gen
erally unknown in this country to the present day is a somewhat
remarkable fact ; because the name of the author, learned with
the very elements of science, is revered at every hearth-stone
where knowledge and virtue are of chief esteem, while, abroad,
in all the high places of the land, the character which that name
recalls is held up as the noblest illustration of what MAN may be,
and may do, in the possession and manifestation of pre-eminent
intellectual and moral worth ; because the work is celebrated, not
only in the history of one career and one mind, but in the history
of all achievement and human reason itself; because of the spirit
of inquiry, which has been aroused, and which, in pursuing its
searchings, is not always satisfied with stopping short of the foun
tain-head of any given truth ; and, finally, because of the earnest
endeavour that has been and is constantly going on, in many
sections of the Republic, to elevate the popular standard of edu
cation and give to scientific and other efforts a higher and a
better aim.

True, the PRINCIPIA has been hitherto inaccessible to popular
use. A few copies in Latin, and occasionally one in English may
be found in some of our larger libraries, or in the possession of
some ardent disciple of the great Master. But a d^ad language
in the one case, and an enormous price in both, particularly in
that of the English edition, have thus far opposed very sufficient
obstacles to the wide circulation of the work. It is now, how
ever, placed within the reach of all. And in performing this la
bour, the utmost care has been taken, by collation, revision, and
otherwise, to render the First American Edition the most accurate
and beautiful in our language. u Le plus beau monument que
l? on puisse clever a la gloire de Newton, c'est une bonne edition
de ses ouvrages :" and a monument like unto that we would here

V: INTRODUCTION TO

set up. The PRINCIPIA, above all, glows with the immortality of
a transcendant mind. Marble and brass dissolve and pass away ;
but the true creations of genius endure, in time and beyond time,
forever : high upon the adamant of the indestructible, they send
forth afar and near, over the troublous waters of life, a pure, un
wavering, quenchless light whereby the myriad myriads of barques,
richly laden with reason, intelligence and various faculty, are
guided through the night and the storm, by the beetling shore
and the hidden rock, the breaker and the shoal, safely into havens
calm and secure.

To the teacher and the taught, the scholar and the student, the
devotee of Science and the worshipper of Truth, the PRINCIPIA
must ever continue to be of inestimable value. If to educate
means, not so much to store the memory with symbols and facts,
as to bring forth the faculties of the soul and develope them to the
full by healthy nurture and a hardy discipline, then, what so effec
tive to the accomplishment of that end as the study of Geometri
cal Synthesis ? The Calculus, in some shape or other, is, indeed,
necessary to the successful prosecution of researches in the higher
branches of philosophy. But has not the Analytical encroached
upon the Synthetical, and Algorithmic Formulae been employed
when not requisite, either for the evolution of truth, or even its
apter illustration ? To each method belongs, undoubtedly, an
appropriate use. Newton, himself the inventor of Fluxions,
censured the handling of Geometrical subjects by Algebraical
calculations ; and the maturest opinions which he expressed were
additionally in favour of the Geometrical Method. His prefer
ence, so strongly marked, is not to be reckoned a mere matter oi
taste ; and his authority should bear with preponderating weight
upon the decision of every instructor in adopting what may be
deemed the best plan to insure the completes! mental develop
ment. Geometry, the vigorous product of remote time ; blended
with the earliest aspirations of Science and the earliest applica
tions of Art ; as well in the measures of music as in the move
ment of spheres ; as wholly in the structure of the atom as in that
of the world; directing MOTION and shaping APPEARANCE; in a
wonl, *t the moulding of the created all, is, in comprehensive

THE AMERICAN EDITION. Vll

view, the outward form of that Inner Harmony of which and in
which all things are. Plainly, therefore, this noble study has
other and infinitely higher uses than to increase the power of ab
straction. A more general and thorough cultivation of it should
oe strenuously insisted on. Passing from the pages of Euclid or
Legendre, might not the student be led, at the suitable time, to
those of the PRINCIPIA wherein Geometry may be found in varied
use from the familiar to the sublime ? The profoundest and the
happiest results, it is believed, would attend upon this enlargement
of our Educational System.

Let the PRINCIPIA, then, be gladly welcomed into every Hall
where a TRUE TEACHER presides. And they who are guided to
the diligent study of this incomparable work, who become
strengthened by its reason, assured by its evidence, and enlight
ened by its truths, and who rise into loving communion with the
great and pure spirit of its author, will go forth from the scenes
of their pupilage, and take their places in the world as strong-
minded, right-hearted men — such men as the Theory of our
Government contemplates and its practical operation absolutely
demands.

LIFE OF

SIE ISAAC NEWTON.

Nec fas est proprius mortal? attingere Divos. — HALLEY.

FROM the thick darkness of the middle ages man's struggling
spirit emerged as in new birth ; breaking out of the iron control
of that period ; growing strong and confident in the tug and din
of succeeding conflict and revolution, it bounded forwards and
upwards with resistless vigour to the investigation of physical and
moral truth ; ascending height after height ; sweeping afar over
the earth, penetrating afar up into the heavens ; increasing in en
deavour, enlarging in endowment ; every where boldly, earnestly
out-stretching, till, in the AUTHOR of the PRINCIPIA, one arose,
who, grasping the master-key of the universe and treading its
celestial paths, opened up to the human intellect the stupendous
realities of the material world, and, in the unrolling of its harmo
nies, gave to the human heart a new song to the goodness, wis
dom, and majesty of the all-creating, all-sustaining, all-perfect
God.

Sir Isaac Newton, in whom the rising intellect seemed to attain,
as it were, to its culminating point, was born on the 25th of De
cember, O. S. 1642 — Christmas day — at Woolsthorpe, in the
parish of Colsterworth, in Lincolnshire. His father, John New
ton, died at the age of thirty-six, and only a few months after his
marriage to Harriet Ayscough, daughter of James Ayscough, oi
Rutlandshire. Mrs. Newton, probably wrought upon by the
early loss of her husband, gave premature birth to her only and
posthumous child, of which, too, from its extreme diminutiveness,
she appeared likely to be soon bereft. Happily, it was otherwise
decreed ! The tiny infant, on whose little lips the breath of life

10 LIFE OF SIR ISAAC NEWTON.

so doubtingly hovered, lived ; — lived to a vigorous maturity, to a
hale old age ; — lived to become the boast of his country, the won
der of his time, and the "ornament of his srjecies."

Beyond the grandfather, Robert Newton, the descent of Sir
Isaac cannot with certainty be traced. Two traditions were held
in the family : one, that they were of Scotch extraction ; the
other, that they came originally from Newton, in Lancashire,
dwelling, for a time, however, at Westby, county of Lincoln, be
fore the removal to and purchase of Woolsthorpe — about a hundred
years before this memorable birth.

The widow Newton was left with the simple means of a com
fortable subsistence. The Woolsthorpe estate together with
small one which she possessed at Sewstern, in Leicestershire, yield
ed her an income of some eighty pounds ; and upon this limited sum,
she had to rely chiefly for the support of herself, and the educa
tion of her child. She continued his nurture for three years,
when, marrying again, she confided the tender charge to the care
of her own mother.

Great genius is seldom marked by precocious development ;
and young Isaac, sent, at the usual age, to two day schools at
Skillington and Stoke, exhibited no unusual traits of character.
In his twelfth year, he was placed at the public school at Gran-
tham, and boarded at the house of Mr. Clark, an apothecary.
But even in this excellent seminary, his mental acquisitions con
tinued for a while unpromising enough : study apparentlv had no
charms for him ; he was very inattentive, and ranked low in the
school. One day, however, the boy immediately above our seem
ingly dull student gave him a severe kick in the stomach ; Isaac,
deeply affected, but with no outburst of passion, betook himself,
with quiet, incessant toil, to his books ; he quickly passed above
the offending classmate ; yet there he stopped not ; the strong
spirit was, for once and forever, awakened, and, yielding to itb
noble impulse, he speedily took up his position at the head of all.

His peculiar character began now rapidly to unfold itself.
Close application grew to be habitual. Observation alternated
with reflection. " A sober, silent, thinking lad," yet, the wisest
and the kindliest, the indisputable leader of his fellows. Gener-

LIFE OF SIR ISA VC NEWTON. 11

osity, modesty, and a love of truth distinguished him then as ever
afterwards. He did not often join his classmates in play ; but he
would contrive for them various amusements of a scientific kind.
Paper kites he introduced ; carefully determining their best form
and proportions, and the position and number of points whereby
to attach the string. He also invented paper lanterns ; these
served ordinarily to guide the way to school in winter mornings,
but occasionally for quite another purpose ; they were attached to
the tails of kites in a dark night, to the dismay of the country people
dreading portentous comets, and to the immeasureable delight ol
his companions. To him, however, young as he was, life seemed
to have become an earnest thing. When not occupied with his
studies, his mind would be engrossed with mechanical contrivances ;
now imitating, now inventing. He became singularly skilful in the
use of his little saws, hatchets, hammers, and other tools. A
windmill was erected near Grantham ; during the operations ol
the workmen, he was frequently present ; in a short time, he had
completed a perfect working model of it, which elicited general
admiration. Not content, however, with this exact imitation, he
conceived the idea of employing, in the place of sails, animal power ,
and, adapting the construction of his mill accordingly, he enclosed
in it a mouse, called the miller, and which by acting on a sort ot
treadvvheel, gave motion to the machine. He invented, too, a
mechanical carriage — having four wheels, and put in motion with
a handle worked by the person sitting inside. The measurement
of time early drew his attention. He h'rst constructed a water
clock, in proportions somewhat like an old-fashioned house clock.
The index of the dial plate was turned by a piece of wood acted
upon by dropping water. This instrument, though long used by
himself, and by Mr. Clark's family, did not satisfy his inquiring
mind. His thoughts rose to the sun ; and, by careful and oft-re
peated observations of the solar movements, he subsequently
formed many dials. One of these, named Isaac's dial, was the
accurate result of years' labour, and was frequently referred to
for the hour of the day by the country people.

May we not discern in these continual efforts — the diligent re
search^ the patient meditation, the aspiring glance, and the energy

12 LIFE OF SIR ISAAC NEWTON.

of discovery — the stirring elements of that wondrous spirit,
which, clear, calm, and great, moved, in after years, through
deep onward through deep of Nature's mysteries, unlocking her
strongholds, dispelling darkness, educing order — everywhere si
lently conquering.

Newton had an early and decided taste for drawing. Pictures,
taken sometimes from copies, but often from life, and drawn,
coloured and framed by himself, ornamented his apartment. He
was skilled also, in poetical composition, " excelled in making
verses ;" some of these were borne in remembrance and repeated,
seventy years afterward, by Mrs. Vincent, for whom, in early
youth, as Miss Storey, he formed an ardent attachment. She
was the sister of a physician resident near Woolsthorpe ; but
Newton's intimate acquaintance with her began at Grantham.
where they were both numbered among the inmates of the same
house. Two or three years younger than himself, of great per
sonal beauty, and unusual talent, her society afforded him the
greatest pleasure ; and their youthful friendship, it is believed,
gradually rose to a higher passion ; but inadequacy of fortune
prevented their union. Miss Storey was afterwards twice mar
ried ; Newton, never; his esteem for her continued unabated
during life, accompanied by numerous acts of attention and
kindness.

In 1656, Newton's mother was again left a widowr, and took
up her abode once more at Woolsthorpe. He was now fifteen
years of age, and had made great progress in his studies ; but she,
desirous of his help, and from motives of economy, recalled him
from school. Business occupations, however, and the manage
ment of the farm, proved utterly distasteful to him. When sent to
Grantham Market on Saturdays, he would betake himself to his
former lodgings in the apothecary's garret, where some of Mr.
Clark's old books employed his thoughts till the aged and trust
worthy servant had executed the family commissions and announced
the necessity of return : or, at other times, our young philosopher
would seat himself under a hedge, by the wayside, and continue
his studies till the same faithful personage — proceeding alone to
the town and completing the day's business — stopped as he re-

LIFE OF SIR ISAAC NEWTON, 13

turned. The more immediate affairs of the farm received no
better attention. In fact, his passion for study grew daily more
absorbing, and his dislike for every other occupation more in
tense. His mother, therefore, wisely resolved to give him all the
advantages which an education could confer. He was sent back
to Grantham school, where he remained for some months in busy
preparation for his academical studies. At the recommendation
of one of his uncles, who had himself studied at Trinity College,
Cambridge, Newton proceeded thither, and was duly admitted.
on the 5th day of June 1660, in the eighteenth year of his age.

The eager student had now entered upon a new and wider
field ; and we find him devoting himself to the pursuit of know
ledge with amazing ardour and perseverance. Among other sub
jects, his attention was soon drawn to that of Judicial Astrology
He exposed the folly of this pseudo-science by erecting a figure
with the aid of one or two of the problems of Euclid ; — and thus
began his study of the Mathematics. His researches into this
science were prosecuted with unparallelled vigour and success.
Regarding the propositions contained in Euclid as self-evident
truths, he passed rapidly over this ancient system — a step which
he afterward much regretted — and mastered, without further pre
paratory study, the Analytical Geometry of Descartes. Wallis's
Arithmetic of Infinites, Saunderson's Logic, and the Optics of
Kepler, he also studied with great care ; writing upon them
many comments ; and, in these notes on Wallis's work was un
doubtedly the germ of his fluxionary calculus. His progress was
so great that he found himself more profoundly versed than his tutor
in many branches of learning. Yet his acquisitions were not
gotten with the rapidity of intuition ; but they were thoroughly
made and firmly secured. Quickness of apprehension, or Intel
lectual nimbleness did not belong to him. He saw too far : his,
insight was too deep. He dwelt fully, cautiously upon the least
subject ; while to the consideration of the greatest, he brought a
massive strength joined with a matchless clearness, that, regard
less of the merely trivial or unimportant, bore with unerring sa
gacity upon the prominences of the subject, and, grappling with
its difficulties, rarely failed to surmount them.

14 LIFE OF SIR ISAAC NEWTON

His early and fast friend, Dr. Barrow — in compass of inven
tion only inferior to Newton — who had been elected Professor
of Greek in the University, in 1660, was made Lucasian Profes
sor of Mathematics in 1663, and soon afterward delivered his
Optical Lectures : the manuscripts of these were revised by New
ton, and several oversights corrected, and many important sug
gestions made by him ; but they were not published till 1669.

In the year 1665, he received the degree of Bachelor of Arts ;
and, in 1666, he entered upon those brilliant and imposing dis
coveries which have conferred inappreciable benefits upon science,
and immortality upon his own name.

Newton, himself, states that he was in possession of his Method
of Fluxions, " in the year 1666, or before." Infinite quantities
had long been a subject of profound investigation ; among the
ancients by Archimedes, and Pappus of Alexandria ; among the
moderns by Kepler, Cavaleri, Roberval, Fermat and Wallis.
With consummate ability Dr. Wallis had improved upon the la-
hours of his predecessors : with a higher power, Newton moved
forwards from where Wallis stopped. Our author first invented
his celebrated BINOMIAL THEOREM. And then, applying this
Theorem to the rectification of curves, and to the determination
of the surfaces and contents of solids, and the position of their
centres of gravity, he discovered the general principle of deducing
the areas of curves from the ordinate, by considering the area as
a nascent quantity, increasing by continual fluxion in the propor
tion of the length of the ordinate, and supposing the abscissa
to increase uniformly in proportion to the time. Regarding lines
as generated by the motion of points, surfaces by the motion of
lines, and solids by the motion of surfaces, and considering that
the ordinates, abscissae, &c., of curves thus formed, vary accord
ing to a regular law depending on the equation of the curve,
he deduced from this equation the velocities with which these
quantities are generated, and obtained by the rules of infinite
series, the ultimate value required. To the velocities with which
every line or quantity is generated, he gave the name of FLUX
IONS, and to the lines or quantities themselves, that of FLUENTS.
A discovery that successively baffled the acutest and strongest

LIFE OF SIR ISAAC NEWTON. 15

intellects : — that, variously modified, has proved of incalculable
service in aiding to develope the most abstruse and the highest
'ruths in Mathematics and Astronomy : and that was of itself
enough to render any name illustrious in the crowded Annals of
Science.

At this period, the most distinguished philosophers were direct
ing all their energies to the subject of light and the improvement
of the refracting telescope. Newton, having applied himself to
the grinding of "optic glasses of other figures than spherical," ex
perienced the impracticability of executing such lenses ; and con
jectured that their defects, and consequently those of refracting
telescopes, might arise from some other cause than the imperfect
convergency of rays to a single point. He accordingly "procured
a triangular glass prism to try therewith the celebrated phenom
ena of colours." His experiments, entered upon with zeal, and
conducted with that industry, accuracy, and patient thought, for
which he was so remarkable, resulted in the grand conclusion,
that LIGHT WAS NOT HOMOGENEOUS, BUT CONSISTED OF RAYS,

SOME OF WHICH WERE MORE REFRANGIBLE THAN OTHERS. This

profound and beautiful discovery opened up a new era in the
History of Optics. As bearing, however, directly upon the construc
tion of telescopes, he saw that a lens refracting exactly like a prism
would necessarily bring the different rays to different foci, at
different distances from the glass, confusing and rendering the
vision indistinct. Taking for granted that all bodies produced
spectra of ^ jtial length, he dismissed all further consideration of
the refracting instrument, and took up the principle of reflection.
Rays of all colours, he found, were reflected regularly, so that the
angle of reflection was equal to the angle of incidence, and hence
he concluded that ojitical instruments might be brought to any
degree of perfection imaginable, provided reflecting specula of
the requisite figure and finish could be obtained. At this stage
of his optical researches, he was forced to leave Cambridge on
account of the plague which was then desolating England.

He retired to Woolsthorpe. The old manor-house, in which he
was born, was situated in a beautiful little valley, on the west side
of the river Witham ; and here in the quiet home of his boyhood,

2

16 LIFE OF SIR ISAAC NEWTON.

he passed his days in serene contemplation, while the stalking
pestilence was hurrying its tens of thousands into undistinguisha •
ble graves.

Towards the close of a pleasant day in the early autumn of
1666, he was seated alone beneath a tree, in his garden, absorbed
in meditation. He was a slight young man ; in the twenty-fourth
year of his age ; his countenance mild and full of thought. For
a century previous, the science of Astronomy had advanced with
rapid strides. The human mind had risen from the gloom and
bondage of the middle ages, in unparalleled vigour, to unfold the
system, to investigate the phenomena, and to establish the laws
of the heavenly bodies. Copernicus, Tycho Brahe, Kepler,
Galileo, and others had prepared and lighted the way for him
who was ta give to their labour its just value, and to their genius
its true lustre. At his bidding isolated facts were to take order
as parts of one harmonious whole, and sagacious conjectures grow
luminous in the certain splendour of demonstrated truth. And
this ablest man had come — was here. His mind, familiar with
the knowledge of past effort, and its unequalled faculties develop
ed in transcendant strength, was now moving on to the very
threshold of Its grandest achievement. Step by step the untrod
den path was measured, till, at length, the entrance seemed dis
closed, and the tireless explorer to stand amid the first opening
wonders of the universe.

The nature of gravity — that mysterious power which causes
all bodies to descend towards the centre of the earth — had, in
deed, dawned upon him. And reason busily united link to link
of that chain which was yet to be traced joining the least to the
vastest, the most remote to the nearest, in one harmonious bond.
From the bottoms of the deepest caverns to the summits of the
highest mountains, this power suffers no sensible change : may not
its action, then, extend to the moon ? Undoubtedly : and furthei
reflection convinced him that such a power might be .sufficient for
retaining that luminary in her orbit round the earth. But, though
this power suffers no sensible variation, in the little change of
distance from the earth's centre, at which we may place our-
.'«lves, yet, at the distance of the moon, :miy not its force undergo

LIFE OF SIR ISAAC NEWTON. 17

more or less diminution ? The conjecture appeared most proba
ble : and, in order to estimate what the degree of diminution
might be, he considered that if the moon be retained in her orbit
by the force of gravity, the primary planets must also be carried
round the sun by the like power; and, by comparing the periods
of the several planets with their distances from the sun, he found
that, if they were held in their courses by any power like gravity,
its strength must decrease in the duplicate proportion of the in
crease of distance. In forming this conclusion, he supposed the
planets to move in perfect circles, concentric to the sun. Now
was this the law of the moon's motion ? Was such a force, em
anating from the earth and directed to the moon, sufficient, when
diminished as the square of the distance, to retain her in her
orbit ? To ascertain this master-fact, he compared the space
through which heavy bodies fall, in a second of time, at a given
distance from the centre of the earth, namely, at its surface, with
the space through which the moon falls, as it were, to the earth,
in the same time, while revolving in a circular orbit. He was
absent from books ; and, therefore, adopted, in computing the
earth's diameter, the common estimate of sixty miles to a degree
of latitude as then in use among geographers and navigators.
The result of his calculations did not, ot course, answer his ex
pectations ; hence, he concluded that some other cause, beyond the
reach of observation — analogous, perhaps, to the vortices of Des
cartes — joined its action to that of the power of gravity upon the
rnooil. Though by no means satisfied, he yet abandoned awhile
further inquiry, and remained totally silent upon the subject.

These rapid marches in the career of discovery, combined with
the youth of Newton, seem to evince a penetration the most
lively, and an invention the most exuberant. But in him there
was a conjunction of influences as extraordinary as fortunate.
Study, unbroken, persevering and profound carried on its inform
ing and disciplining work upon a genius, natively the greatest,
and rendered freest in its movements, and clearest in its vision,
through the untrammelling and enlig} tenirig power of religion.
And, in this happy concurrence, are to be sought the elements of
those amazing abilities, which, grasping, with equal facility, the

18 LIFE OF SIR ISAAC NEWTON.

minute and the stupendous, brought these successively to light,
and caused science to make them her own.

In 1667, Newton was made a Junior Fellow ; and, in the year
following, he took his degree of Master of Arts, and was appoint
ed to a Senior Fellowship.

On his return to Cambridge, in 1668, he resumed his optical
labours. Having thought of a delicate method of polishing metal,
he proceeded to the construction of his newly projected reflect
ing telescope ; a small specimen of which he actually made with
his own hands, It was six inches long ; and magnified about
forty times ; — a power greater than a refracting instrument of six
feet tube could exert with distinctness. Jupiter, with his four
satellites, and the horns, or moon-like phases of Venus were
plainly visible through it. THIS WAS THE FIRST REFLECTING

TELESCOPE EVER EXECUTED AND DIRECTED TO THE HEAVENS.

He gave an account of it, in a letter to a friend, dated February 23d,
1668-9 — a letter which is also remarkable for containing the firs'
allusion to his discoveries " concerning the nature of light." En
couraged by the success of his first experiment, he again executed
with his own hands, not long afterward, a second and superior
instrument of the same kind. The existence of this having come
to the knowledge of the Royal Society of London, in 1671, they
requested it of Newton for examination. He accordingly sent it
to them, It excited great admiration; it was shown to the king*
a drawing and description of it was sent to Paris ; and the tele-
scope itself was carefully preserved in the Library of the Society.
Newton lived to see his invention in public use, and of eminent
service in the cause of science.

In the spring of 1669, he wrote to his friend Francis Aston,
Esq., then about setting out on his travels, a letter of advice and
directions, it was dated May 18th, and is interesting as exhibit
ing some of the prominent features in Newton's character.
Thus : —

" Since in your letter you give me so much liberty of spending
my judgment about what may be to your advantage in travelling,
1 shall do it more freely than perhaps otherwise would have been
decent, Fir,ct, then, I will lay down some general rules, most of

LIFE OF SIR ISAAC NEWTON. 19

which, I bolieA e, you have considered already ; but if any of
them be new to you, they may excuse the rest ; if none at all,
yet is my punishment more in writing than yours in reading.

"When you come into any fresh company. 1. Observe their
humours. 2. Suit your own carriage thereto, by which insinua
tion you will make their converse more free and open. 3. Let
your discourse be more in queries and doubtings than peremptory
assertions or disputings, it being the design of travellers to learn,
not to teach. Besides, it will persuade your acquaintance that
you have the greater esteem of them, and so make them more
ready to communicate what they know to you ; whereas nothing
sooner occasions disrespect and quarrels than peremptoriness.
You will find little or no advantage in seeming wiser or much
more ignorant than your company. 4. Seldom discommend any
thing though never so bad, or do it but moderately, lest you be
unexpectedly forced to an unhandsome retraction. It is safer to
commend any thing more than it deserves, than to discommend
a thing so much as it deserves; for commendations meet not
so often with oppositions, or, at least, are not usually so ill re
sented by men that think otherwise, as discommendations ; and
you will insinuate into men's favour by nothing sooner than seem
ing to approve and commend what they like ; but beware o
doing it by comparison. 5. If you be affronted, it is better, in c
foreign country, to pass it by in silence, and with a jest, though
with some dishonour, than to endeavour revenge ; for, in the first
case, your credit's ne'er the worse when you return into England,
or come into other company that have not heard of the quarrel.
But, in the second case, you may bear the marks of the quarrel
while you live, if you outlive it at all. But, if you find yoursell
unavoidably engaged, 'tis best, I think, if you can command your
passion and language, to keep them pretty evenly at some certain
moderate pitch, not much heightening them to exasperate your
adversary, or provoke his friends, nor letting them grow overmuch
dejected to make him insult. In a word, if you can keep reason
above passion, that and watchfulness will be your best defendants.
To which purpose you may consider, that, though such excuses
is this — He provok't me so much I could not forbear — may pass

20 LIFE OF SIR ISAAC NEWTON.

among friends, yet amongst strangers they are insignificant, ina
only argue a traveller's weakness.

" To these I may add some general heads for inquiries or ob
servations, such as at present I can think on. As, 1. To observe
the policies, wealth, and state affairs of nations, so far as a soli-
fary traveller may conveniently do. 2. Their impositions upon
all sorts of people, trades, or commodities, that are remarkable.
3. Their laws and customs, how far they differ from ours. 4.
Their trades and arts wherein they excel or come short of us in
England. 5. Such fortifications as you shall meet with, their
fashion, strength, and advantages for defence, and other such mili
tary affairs as are considerable. 6. The power and respect be
longing to their degrees of nobility or magistracy. 7. It will not
be time misspent to make a catalogue of the names and excellen
cies of those men that are most wise, learned, or esteemed in any
nation. 8. Observe the mechanism and manner of guiding ships.

9. Observe the products of Nature in several places, especially in
mines, with the circumstances of mining and of extracting metals
or minerals out of their ore, and of refining them ; and if you
meet with any transmutations out of their own species into
another (as out of iron into copper, out of any metal into quick
silver, out of one salt into another, or into an insipid body, &c.),
those, above all, will be worth your noting, being the most lucif-
erous, and many times lucriferous experiments, too, in philosophy.

10. The prices of diet and other things. 11. And the staple
commodities of places.

" These generals (such as at present I could think of), if they
will serve for nothing else, yet they may assist you in drawing up
a model to regulate your travels by. ' As for particulars, these that
follow are all that 1 can now think of, viz. ; whether at Schem-
nitium, in Hungary (where there are mines of gold, copper, iron,
vitriol, antimony, &c.). they change iron into copper by dissolving
»t in a vitriolate water, which they find in cavities of rocks in the
mines, and then melting the slimy solution in a stroi ig fire, which
in the cooling proves copper. The like is said to be done in other
places, which I cannot now remember ; perhaps, too, it may be
lone in Italy. For about twenty or thirty years agone there was

LIFE OF SIR ISAAC NEWTON, 21

a certain vitriol came from thence (called Roman vitriol), but of
a nobler virtue than that which is now called by that name ;
which vitriol is not now to be gotten, because, perhaps, they make
a greater gain by some such trick as turning iron into copper
with it than by selling it. 2. Whether, in Hungary, Sclavonia,
Bohemia, near the town Eila, or at the mountains of Bohemia
near Silesia, there be rivers whose waters are impregnated with
gold ; perhaps, the gold being dissolved by some corrosive water
like aqua regis, and the solution carried along with the stream,
that runs through the mines. And whether the practice of laying
mercury in the rivers, till it be tinged with gold, and then strain
ing the mercury through leather, that the gold may stay behind,
be a secret yet, or openly practised. 3. There is newly con
trived, in Holland, a mill to grind glasses plane withal, and I
think polishing them too ; perhaps it will be worth the while to see

it. 4. There is in Holland one Borry, who some years since

was imprisoned by the Pope, to have extorted from him secrets
(as I am told) of great worth, both as to medicine and profit, but
he escaped into Holland, where they have granted him a guard.
I think he usually goes clothed in green. Pray inquire what you
can of him, and whether his ingenuity be any profit to the Dutch.
You may inform yourself whether the Dutch have any tricks to
keep their ships from being all worm-eaten in their voyages to
the Indies. Whether pendulum clocks do any service in finding
out the longitude, &c.

" I am very weary, and shall not stay to part with a long
compliment, only I wish you a good journey, and God be with
you."

It was not till the month of June, 1669, that our author made
known his Method of Fluxions. He then communicated the
work which he had composed upon the subject, and entitled,
ANALYSIS PER EQUATIONES NUMERO TERMINORUM INFINITAS,
to his friend Dr. Barrow. The latter, in a letter dated 20th of the
same month, mentioned it to Mr. Collins, and transmitted it to
him, on the 31st of July thereafter. Mr. Collins greatly approv>
ed of the work ; took a copy of it ; and sent the original back
to Dr. Barrow. During the same and the two following years, Mr

< LIFE OF SIR ISAAC NEWTON.

Collins, by his extensive correspondence, spread the knowledge
of this discovery among the mathematicians in England, Scotland,
France, Holland and Italy.

Dr. Barrow, having resolved to devote himself to Theology,
resigned the Lucasian Professorship of Mathematics, in 1669, in
favour of Newton, who accordingly received the appointment to
the vacant chair.

During the years 1669, 1670, and 1671, our author, as such
Professor, delivered a course of Optical Lectures. Though these
contained his principal discoveries relative to the different re-
frangibility of light, yet the discoveries themselves did not be
come publicly known, it seems, till he communicated them to the
Royal Society, a few weeks after being elected a member there
of, in the spring of 1671-2. He now rose rapidly in reputation,
and was soon regarded as foremost among the philosophers of the
age. His paper on light excited the deepest interest in the Royal
Society, who manifested an anxious solicitude to secure the author
from the " arrogations of others," and proposed to publish his
discourse in the monthly numbers in which the Transactions were
given to the world. Newton, gratefully sensible of these expres
sions of esteem, willingly accepted of the proposal for publication.
He gave them also, at this time, the results of some further ex
periments in the decomposition and re-composition of light : — that
the same degree of refrangibility always belonged to the same
colour, and the same colour to the same degree of refrangibility :
that the seven different colours of the spectrum were original, or
simple, and that whiteness^ or white light was a compound of all
these seven colours.

The publication of his new doctrines on light soon called forth
violent opposition as to their soundness. Hooke and Huygens —
men eminent for ability and learning — were the most conspicuous
of the assailants. And though Newton effectually silenced all his
adversaries, yet he felt the triumph of little gain in comparison
.vith the loss his tranquillity had sustained. He subsequently re-
narked in allusion to this controversy — and to one with whom
he was destined to have a longer and a bitterer conflict — " I was
so persecuted with discussions arising from the publication of m v

LIFE OF SIR ISAAC NEWTON. 23

theory ot light, that I blamed my own imprudence for parting
with so substantial a blessing as rny quiet to run after a shadow.7'

In a communication to Mr. Oldenburg, Secretary of the Royal
Society, in 1672, our author stated many valuable suggestions re
lative to the construction of REFLECTING MICROSCOPES which he
considered even more capable of improvement than telescopes.
He also contemplated, about the same time, an edition of Kirick-
huysen's Algebra, with notes and additions; partially arranging,
as an introduction to the work, a treatise, entitled, A Method of
Fluxions ; but he finally abandoned the design. This treatise,
however, he resolved, or rather consented, at a late period of his
life, to put forth separately ; and the plan would probably have
been carried into execution had riot his death intervened. It was
translated into English, and published in 1736 by John Colson,
Professor of Mathematics in Cambridge.

Newton, it is thought, made his discoveries concerning the
INFLECTION and DIFFRACTION of light before 1674. The phe
nomena of the inflection of light had been first discovered more
than ten years before by Grimaldi. And Newton began by re
peating one of the experiments of the learned Jesuit — admitting
a beam of the sun's light through a small pin hole into a dark
chamber : the light diverged from the aperture in the form of a,
cone, and the shadows of all bodies placed in this light were
larger than might have been expected, and surrounded with three
coloured fringes, the nearest being widest, and the most remote
the narrowest. Newton, advancing upon this experiment, took
exact measures of the diameter of the shadow of a human hair,
and of the breadth of the fringes, at different distances behind it,
and discovered that these diameters and breadths were not pro
portional to the distances at which they were measured. He
hence supposed that the rays which passed by the edge of the
hair were deflected or turned aside from it, as if by a repulsive
force, the nearest rays suffering the greatest, the more remote a
less degree of deflection. In explanation of the coloured fringes,
he queried : whether the rays which differ in refrangibility do not
differ also in flexibility, and whether they are n<»t, by these dif
ferent inflections, separated from one another, so as after separa-

< LIFE OF SIR ISAAC NEWTON.

tion to make the colours in the three fringes above described ?
Also, whether the rays, in passing by the edges and sides ol
bodies, are not bent several times backwards and forwards with
an eel-like motion — the three fringes arising from three such
bendings ? His inquiries on this subject were here interrupted
and fiever renewed.

His Theory of the COLOURS of NATURAL BODIES was commu
nicated to the Royal Society, in February, 1675. This is justly
regarded as one of the profoundest of his speculations. The fun
damental principles of the Theory in brief, are : — That bodies
possessing the greatest refractive powers reflect the greatest
quantity of light ; and that, at the confines of equally refracting
media, there is no reflection. That the minutest particles of al
most all natural bodies are in some degree transparent. That
between the particles of bodies there are pores, or spaces, either
empty or filled with media of a less density than the particles
themselves. That these particles, and pores or spaces, have some
definite size. Hence he deduced the Transparency, Opacity, and
colours of natural bodies. Transparency arises from the particles
and their pores being too small to cause reflection at their com
mon surfaces — the light all passing through ; Opacity from the
opposite cause of the particles and their pores being sufficiently
large to reflect the light which is " stopped or stifled7' by the
multitude of reflections ; and colours from the particles, accord
ing to their several sizes, reflecting rays of one colour and trans
mitting those of another — or in other words, the colour that
meets the eye is the colour reflected, while all the other rays are
transmitted or absorbed.

Analogous in origin to the colours of natural bodies, he con
sidered the COLOURS OF THIN PLATES. This subject was interest
ing and important, and had attracted considerable investigation.
He, however, was the first to determine the law of the produc
tion of these colours, arid, during the same year made known the
results of his researches herein to the Royal Society. His mode
of procedure in these experiments was simple and curious. He
placed a double convex lens of a large known radius of curvature,
the flat surface of a plano-convex object glass. Thus, from

UFE OF SIR ISAAC NEWTON. 25

their point of contact at the centre, to the circumference of the
lens, he obtained plates of air, or spaces varying from the ex-
tremest possible thinness, by slow degrees, to a considerable thick
ness. Letting the light fall, every different thickness of this
plate of air gave different colours — the point of contact of the
lens and glass forming the centre of numerous concentric colored
nags. Now the radius of curvature of the lens being known, the
thickness of the plate of air, at any given point, or where any par
ticular colour appeared, could be exactly determined. Carefully
noting, therefore, the order in which the different colours ap
peared, he measured, with the nicest accuracy, the different thick*
nesses at which the most luminous parts of the rings were pro
duced, whether the medium were air, water, or mica — all these
substances giving the same colours at different thicknesses ; — the
ratio of which he also ascertained. From the phenomena obser
ved in these experiments, Newton deduced his Theory of Fits of
EASY REFLECTION AND TRANSMISSION of light. It consists in suppos
ing that every particle of light, from its first discharge from a lumi
nous body, possesses, at equally distant intervals, dispositions to
be reflected from, or transmitted through the surfaces of bodies
upon which it may fall. For instance, if the rays are in a Fit of
Easy Reflection, they are on reaching the surface, repelled,
thrown off] or reflected from it ; if, in a Fit of Easy Transmission,
they are attracted, drawn in, or transmitted through it. By this
Theory of Fits, our author likewise explained the colours of
thick plates.

He regarded light as consisting of small material particles
emitted from shining substances. He thought that these parti
cles could be re-combined into solid matter, so that " gross bodies
and light, were convertible into one another ;" that the particles of
light and the particles of solid bodies acted mutually upon each
other ; those of light agitating and heating those of solid bodies,
and the latter attracting and repelling the former. Newton was
the first to suggest the idea of the POLARIZATION of light.

In the paper entitled An Hypothesis Explaining Properties of
Light, December, 1675, our author first introduced his opinions re
specting Ether — opinions which he afterward abandoned and again

26 LIFE OF SIR S.\AC 1SEWTON.

permanently resumed — " A most subtle spirit which pervades" ah
bodies, and is expanded through all the heavens. It is electric,
and almost, if not quite immeasurably elastic and rare. " By the
force and action of which spirit the particles of bodies mutually
attract one another, at near distances, and cohere, if contiguous ;
and electric bodies operate at greater distances, as well repelling
as attracting the neighbouring corpuscles ; and light is emitted,
-reflected, refracted, inflected and heats bodies ; and all sensation
is excited, and the members of animal bodies move at the com
mand of the will, namely, by the vibrations of this spirit, mutu
ally propagated along the solid filaments of the nerves, from the
outward organs of sense to the brain, and from the brain into the
muscles." This " spirit" was no anima mundi ; nothing further
from the thought of Newton ; but was it not, on his part, a par
tial recognition of, or attempt to reach an ultimate material force,
or primary element, by means of which, " in the roaring loom of
time," this material universe, God's visible garment, may be
woven for us ?

The Royal Society were greatly interested in the results of
some experiments, which our author had, at the same time, com
municated to them relative to the excitation of electricity in glass ;
and they, after several attempts and further direction from him,
succeeded in re-producing the same phenomena.

One of the most curious of Newton's minor inquiries related to
the connexion between the refractive powers and chemical com
position of bodies. He found on comparing the refractive powers
and the densities of many different substances, that the former
were very nearly proportional to the latter, in the same bodies.
Unctuous and sulphureous bodies were noticed as remarkable excep
tions — as well as the diamond — their refractive powers being two
or three times greater in respect of their densities than in the
case of other substances, while, as among themselves, the one was
generally proportional to the other. He hence inferred as to the
diamond a great degree of combustibility •; — a conjecture which
the experiments of modern chemistry have shown to be true.

The chemical researches of our author were probably pursued
with more or less diligence from the time of his witnessing some

LIFE OF .SIR ISAAC NEWTON. 27

?t' the uractical operations in that science at the Apothecary's at
Grantham. DE NATURA ACIDORUM is a short chemical paper, on
various topics, and published in Dr. Horsley's Edition of his
works. TABULA QUANTITATUM E r GRADUUM COLORIS was in
serted iii the Philosophical Transactions ; it contains a compara
tive scale of temperature from that of melting ice to that of a
small kitchen coal-fire. He regarded fire as a body heated so hot
as to emit light copiously ; and flame as a vapour, fume, or ex
halation heated so hot as to shine. To elective attraction, by
the operation of which the small particles of bodies, as he con
ceived, act upon one another, at distances so minute as to escape
observation, he ascribed all the various chemical phenomena ot
precipitation, combination, solution, and crystallization, and the
mechanical phenomena of cohesion and capillary attraction. New
ton's chemical views were illustrated and confirmed, in part, at
least, in his own life-time. As to the structure of bodies, he was
of opinion " that the smallest particles of matter may cohere by
the strongest attractions, and compose bigger particles of weaker
virtue ; and many of these may cohere and compose bigger par
tides whose virtue is still weaker ; and so on for divers succes
sions, until the progression end in the biggest particles, on which
the operations in chemistry and the colours of natural bodies de
pend, and which by adhering, compose bodies of sensible magni
tude."

There is good reason to suppose that our author was a diligent
student of the writings of Jacob Behmen ; and that in conjunction
with a relative, Dr. Newton, he was busily engaged, for several
months in the earlier part of life, in quest of the philosopher's
tincture. " Great Alchymist," however, very imperfectly de
scribes the character of Behmen, whose researches into things
material and things spiritual, things human and things divine, ai-
ford the strongest evidence of a great and original mind.

More appropriately here, perhaps, than elsewhere, may be
given Newton's account of some curious experiments, made in his
own person, on the action of light upon the retina, Locke, who
was an intimate friend of our author, wrote to him for his opinion
on a certain fact stated in Boyle's Book of Colours. Newton, in

2S LIFE OF SIR ISAAC NEWTON.

his reply, dated June 30th, 16(Jl, narrates the following circum
stances, which probably took place in the course of his optical
researches. Thus : —

" The observation you mention in Mr. Boyle's Book of Colours
I once tried upon myself with the hazard of my eyes. The
manner was this ; I looked a very little while upon the sun in the
looking-glass with my right eye, and then turned my eyes into a
dark corner of my chamber, arid winked, to observe the impres
sion made, and the circles of colours which encompassed it, and
how they decayed by degrees, and at last vanished. This I re
peated a second and a third time. At the third time, when the
phantasm of light and colours about it were almost vanished, in
tending my fancy upon them to see their last appearance, I found,
to my amazement, that they began to return, and by little and
little to become as lively and vivid as when I had newly looked
upon the sun. But when I ceased to intend my fancy upon them,
they vanished again. After this, I found, that as often as I went
into the dark, and intended my mind upon them, as when a man
looks earnestly to see anything which is difficult to be seen, I
could make the phantasm return without looking any more upon
the sun ; and the oftener I made it return, the more easily I could
make it return again. And, at length, by repeating this, without
looking any more upon the sun, I made such an impression on my
eye, that, if I looked upon the clouds, or a book, or any bright
object, I saw upon it a round bright spot of light like the sun,
and, which is still stranger, though I looked upon the sun with
my right eye only, and not with my left, yet my fancy began *o
make an impression upon my left eye, as well us upon my right.
For if I shut my right eye, or looked upon a book, or the clouds,
with my left eye, I could see the spectrum of the sun almost as
plain as with my right eye, if I did but intend my fancy a little
while upon it ; for at first, if I shut my right eye, and looked with
my left, the spectrum of the sun did not appear till I intended my
fancy upon it ; but by repeating, this appeared every time more
easily. And now, in a few hours' time, I had brought my eyes
to such a pass, that I could look upon no blight object with either
eye, but I saw the sun before me, so that I durst neither write

LIFE OF SIR ISAAC NEWTON. 29

nor read ; but to recover the use of my eyes, shut myself up in
my chamber made dark, for three days together, and used all
means to divert my imagination from the sun. For if I thought
upon him, I presently saw his picture, though I was in the dark.
But by keeping in the dark, and employing my mind about other
things, I began in three or four days to have some use of my eyes
again ; and by forbearing to look upon bright objects, recovered
them pretty well, though not so well but that, for some months
after, the spectrum of the sun began to return as often as I began
to meditate upon the phenomena, even though I lay in bed at mid
night with my curtains drawn. But now I have been very well
for many years, though I am apt to think, if I durst venture my
eyes, I could still make the phantasm return by the power of my
fancy. This story I tell you, to let you understand, thaj; in the
observation related by Mr. Boyle, the man's fancy probably con
curred with the impression made by the sun's light to produce
that phantasm of the sun which he constantly saw in bright ob
jects. And so your question about the cause of phantasm in
volves another about the power of fancy, which I must confess is
too hard a knot for me to untie. To place this effect in a constant
motion is hard, because the sun ought then to appear perpetually.
It seems rather to consist in a disposition of the sensorium to
move the imagination strongly, and to be easily moved, both by
the imagination and by the light, as often as bright objects are
looked upon."J

Though Newton had continued silent, yet his thoughts were
by no means inactive upon the vast subject of the planetary mo
tions. The idea of Universal Gravitation, first caught sight of, so
to speak, in the garden at Woolsthorpe, years ago, had gradually
expanded upon him. We find him, in a letter to Dr. Hooke,
Secretary of the Royal Society, dated in November, 1679, pro
posing to verify the motion of the earth by direct experiment,
namely, by the observation of the path pursued by a body falling
from a considerable height. He had concluded that the path
would be spiral ; but Dr. Hooke maintained that it would be an
eccentric ellipse iu vacuo, and an ellipti-spiral in a resisting me
dium. Our author, aided by this correction of his error, and by

30 LIFE OF SIR ISAAC NEWTON.

the discovery that a projectile would move in an elliptical orbil
when under the influence of a force varying inversely as the
square of the distance, was led to discover " the theorem bj
which he afterwards examined the ellipsis ;" and to demonstrate
the celebrated proposition that a planet acted upon by an attrac
tive force varying inversely as the squares of the distances will
describe an elliptical orbit, in one of whose foci the attractive
force resides.

When he was attending a meeting of the Royal Society, in
June 1682, the conversation fell upon the subject of the measure
ment of a degree of the meridian, executed by M. Picard, a
French Astronomer, in 1679. Newton took a memorandum oi
the result ; and afterward, at the earliest opportunity, computed
from it the diameter of the earth : furnished with these new data,
he resumed his calculation of 1666. As he proceeded therein,
he saw that his early expectations were now likely to be realized ;
the thick rushing, stupendous results overpowered him ; he be
came unable to carry on the process of calculation, and intrusted
its completion to one of his friends. The discoverer had, indeed,
grasped the master-fact. The law of falling bodies at the earth's
surface was at length identified with that which guided the moon
in her orbit. And so his GREAT THOUGHT, that had for sixteen
years loomed up in dim, gigantic outline, amid the first dawn of a
plausible hypothesis, now stood forth, radiant and not less grand,
in the mid-day light of demonstrated truth.

It were difficult, nay impossible to imagine, even, the influence
of a result like this upon a mind like Newton's. It was as if the
keystone had been fitted to the glorious arch by which his spirit
should ascend to the outskirts of infinite space — spanning the immea
surable — weighing the imponderable — computing the incalculable
—mapping out the marchings of the planets, and the far-wander
ings of the comefs, and catching, bring back to earth some clearer
notes of that higher melody which, as a sounding voice, bears
perpetual witness to the design and omnipotence of a creating
Deity.

Newton, extending the law thus obtained, composed a series
of about twelve propositions on the motion of the primary planets

LIFE OF SIR ISAAC NEWTON. 31

about the sun. These were sent to London, and communicated
to the Royal Society about the end of 1683. At or near this pe
riod, other philosophers, as Sir Christopher Wren, Dr. Halley,
and Dr. Hooke, were engaged in investigating the same subject ;
but with no definite or satisfactory results. Dr. Halley, having
seen, it is presumed, our author's propositions, went in August,
1684, to Cambridge to consult with him upon the subject.
Newton assured him that he had brought the demonstration to
perfection. In November, Dr. Halley received a copy of the
work ; and, in the following month^ announced . it to the Royal
Society, with the author's promise to have it entered upon their
Register. Newton, subsequently reminded by the Society of his
promise, proceeded in the diligent preparation of the work, and.
though suffering an interruption of six weeks, transmitted the
manuscript of the first book to London before the end of April.
The work was entitled PHILOSOPHI/E NATURALIS PRINCIPIA
MATHEMATICA, dedicated to the Royal Society, and presented
thereto on the 28th of April, 1685-6. The highest encomiums
were passed upon it ; and the council resolved, on the 19th of
May, to print it at the expense of the Society, and under the di
rection of Dr. Halley. The latter, a few days afterward, com
municated these steps to Newton, who, in a reply, dated the 20th
of June, holds the following language : — " The proof you sent me
I like very well. I designed the whole to consist of three books ;
the second was finished last summer, being short, and only wants
transcribing, and drawing the cuts fairly. Some new propositions
I have since thought on, which I can as well let alone. The
third wants the theory of comets. In autumn last, I spent two
months in calculation to no purpose for want of a good method,
which made me afterward return to the first book, and enlarge it
with diverse propositions, some* relating to comets, others to other
things found ouf last winter. The third I now design to sup
press. Philosophy is such an impertinently litigious lady, that a
man had as good be engaged in l«iw-suits as have to do with her.
I found it so formerly, and now I can no sooner come near her
again, but she gives me warning. The first two books without
the third will not so well bear the title of P/iilosophicc Naturalis

3

32 LIFE OF SIR ISAAC NEWTON.

Principia Mathematicia ; and thereupon I had altered it to this,
De Motu Corporum Libri duo. But after second thought I re
tain the former title. It will help the sale of the book, which I
ought not to diminish now 'tis yours."

This " warning" arose from some pretensions put forth by Dr.
Hooke. And though Newton gave a minute and positive refuta
tions of such claims, yet, to reconcile all differences, he gener
ously added to Prop. IV. Cor. 6, Book I, a Scholium, in which
Wren, Hooke and Halley are acknowledged to have indepen
dently deduced the law of gravity from the second law of
Kepler.

The suppression of the third book Dr. Halley could not endure
to see. " I must again beg you" says he, " not to let your re
sentments run so high as to deprive us of your third book, where
in your applications of your mathematical doctrine to the theory
of comets, and several curious experiments, which, as I guess by
what you write ought to compose it, will undoubtedly render it
acceptable to those who will call themselves philosophers without
mathematics, which are much the greater number." To these
solicitations Newton yielded. There were no "resentments," how
ever, as we conceive, in his " design to suppress." He sought
peace ; for he loved and valued it above all applause. But, in
spite of his efforts for tranquillity's sake, his course of discovery
was all along molested by ignorance or presumptuous rivalry.

The publication of the great work now went rapidly forwards,
The second book was sent to the Society, and presented on the
2d March ; the third, on the 6th April ; and the whole was com
pleted and published in the month of May, 1686-7. In the sec
ond Lemma of the second book, the fundamental principle of his
fiuxionary calculus was, for the first time, given to the world ; but
its algorithm or notation did not appear till published in the
second volume nf Dr. Wallis's works, in 1693.

And thus was ushered into existence The PRINCIPIA — a work
to which pre-eminence above all the productions of the human
intellect has been awarded — a work that must be esteemed of
priceless worth so long as Science has a votary, or a single wor
shipper be left to kneel at the altar of Truth.

LIFE OF SIR ISAAC NEWTON. 33

The entire work bears the general title of THE MATHEMATICAL
PRINCIPLES OF NATURAL PHILOSOPHY. It consists of three books:
the first two, entitled, OF THE MOTION OF BODIES, are occupied
with the laws and conditions of motions and forces, and are illus
trated with many scholia treating of some of the most general
and best established points in philosophy, such as the density and
resistance of bodies, spaces void of matter, and the motion of
sound and light. From these principles, there is deduced, in the
third book, drawn up in as popular a style as possible and entitled,
OF THE SYSTEM OF THE WORLD, the constitution of the system of
i he world. In regard to this book, the author say^ — " I had, indeed,
composed the third Book in a popular method, that it might be read
by many ; but afterwards, considering that such as had not suf-
ficently entered into the principles could not easily discover the
strength of the consequences, nor lay aside the prejudices to which
they had been many years accustomed, therefore, to prevent dis
putes which might be raised upon such accounts, I chose to reduce
the substance of this Book into the form of Propositions (in the
mathematical way), which should be read by those only who had
first made themselves masters of the principles established in the
preceding Books : not that I would advise any one to the previous
study of every Proposition of those Books." — "It is enough it
one carefully reads the Definitions, the Laws of Motion, and the
three first Sections of the first Book. He may then pass on to
this Book, and consult such of the remaining Propositions of the
first two Books, as the references in this, and his occasions shall re
quire." So that " The System of the World" is composed both
" in a popular method," and in the form of mathematical Propo
sitions.

The principle of Universal Gravi' ition, namely, that every
particle of matter is attracted by, or gravitates to, every other
particle of matter, icith a force inversely proportional to the
squares of their distances — is the discovery w? ich characterizes
The PRINCIPIA. This principle the author deduced from the mo
tion of the moon, and the three laws of Kepler — laws, which
Newton, in turn, by his greater law, demonstrated to be true.

From the first law of Kepler, namely, the proportionality of

LIFE OF SIR ISAAC NEWTON.

the areas to t\ie times of their description, our author inferred
that the force which retained the planet in its orbit was always
directed to the sun ; and from the second, namely, that every
planet moves in an ellipse with the sun in one of its foci, he drew
the more general inference that the force by which the planet
moves round that focus varies inversely as the square of its dis
tance therefrom : and he demonstrated that a planet acted upon
by such a force could not move in any other curve than a conic
section ; showing when the moving body would describe a circu
lar, an elliptical, a parabolic, or hyperbolic orbit. He demon
strated, too, that this force, or attracting, gravitating power re
sided in every, the least particle ; but that, in spherical masses, it
operated as if confined to their centres ; so that, one sphere or
body will act upon another sphere or body, with a force directly
proportional to the quantity of matter, and inversely as the square
of the distance between their centres; and that their velocities of
mutual approach will be in the inverse ratio of their quantities o*'
matter. Thus he grandly outlined the Universal Law. Verify
ing its truth by the motions of terrestrial bodies, then by those of
the moon and other secondary orbs, he finally embraced, in one
mighty generalization, the entire Solar System — all the move
ments of all its bodies — planets, satellites and comets — explain
ing and harmonizing the many diverse and theretofore inexplica
ble phenomena.

Guided by the genius of Newton, we see sphere bound to
sphere, body to body, particle to particle, atom to mass, the min
utest part to the stupendous whole — each to each, each to all,
and all to each — in the mysterious bonds of a ceaseless, recipro
cal influence. An influence whose workings are shown to be
alike present in the globular dew-drop, or oblate-spheroidal earth ;
in the falling shower, or vast heaving ocean tides ; in the flying
thistle-down, or fixed, ponderous rock ; in the swinging pendulum,
or time-measuring sun ; in the varying and unequal moon, or
earth's slowly retrograding poles ; in the uncertain meteor, or
oiazing comet wheeling swiftly away on its remote, yet determined
round. An influence, in fine, that may link system to system
through all the star-glowing firmament ; then firmament to iirma-

LIFE OF SIR ISAAC NEWTON. 35

merit ; aye, firmament to firmament, again and again, till, con
verging home, it may be, to some ineffable centre, where more
presently dwells He who inhabiteth immensity, and where infini
tudes meet and eternities have their condux, and where around
move, in softest, swiftest measure, all the countless hosts that
crowd heaven's fathomless deeps.

And yet Newton, amid the loveliness and magnitude of Om
nipotence, lost not sight of the Almighty One. A secondary,
however universal, was not taken for the First Cause. An im
pressed force, however diffused and powerful, assumed not the
functions of the creating, giving Energy. Material beauties,
splendours, and sublimities, however rich in glory, and endless in
extent, concealed not the attributes of an intelligent Supreme.
From the depths of his own soul, through reason and the WORD,
he had risen, a priori, to God : from the heights of Omnipotence,
through the design and law of the builded universe, he proved </
posteriori, a Deity. " I had," says he, " an eye upon such prin
ciples as might work, with considering men, for the belief of a
Deity," in writing the PRINCIPIA ; at the conclusion whereof, he
teaches that — " this most beautiful system of the sun, planets and
comets, could only proceed from the counsel and dominion of an
intelligent and powerful Being. And if the fixed stars are the
centres of other like systems, these, being forme 1 by the like
wise counsels, must be all subject to the dominion of One ; especially
since the light of the fixed stars is of the same nature with the
light of the sun, and from every system light passes into all other
systems : and lest the systems of the fixed stars should, by their
gravity, fall on each other mutually, he hath placed those systems
at immense distances one from another.

" This Being governs all things, not as the soul of the world,
but as Lord over all ; and on account of his dominion he is wont,
to be called Lord God Travrowparwp or Universal Ruler ; for God
is a relative word, and has a respect to servants ; and Deity is
the dominion of God, not over his own body, as those imagine
who fancy God to be the soul of the world, but over servants.
The Supreme God is a Being eternal, infinite, absolutely perfect ;
but a being, however perfect, without dominion, cannot be said to

36 LIFE OF SIR ISAAC NEWTON.

be Lord God ; for we say, my God, your God, the God of Israel
the God of Gods, and Lord of Lords ; but we do not say, my
Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods :
we do not say my Infinite, or my Perfect : these are titles which
have no respect to servants. The word God usually signifies
Lord ; but every Lord is not God. It is the dominion of a spir
itual Being which constitutes a God ; a true, supreme, or imagi
nary dominion makes a true, supreme, or imaginary God. And
from his true dominion it follows that the true God is a living,
intelligent and powerful Being ; and from his other perfections,
that he is supreme or most perfect. He is eternal and in
finite, omnipotent and omniscient ; that is, his duration reaches
from eternity to eternity ; his presence from infinity to infinity ;
he governs all things and knows all things, that are or can be
done. He is not eternity or infinity, but eternal and infinite ;
he is not duration and space, but he endures and is present.
He endures forever and is everywhere present ; and by existing
always and everywhere, he constitutes duration and space. Since
every particle of space is always, and every indivisible moment
of duration is everywhere, certainly the Maker and Lord of things
cannot be never and nowhere. Every soul that has perception
is, though in different times and different organs of sense and mo
tion, still the same indivisible person. There are given succes
sive parts in duration, co-existent parts in space, but neither the
one nor the other in the person of a man, or his thinking
principle ; and much less can they be found in the thinking sub
stance of God. Every man. so far as he is a thing that has j:er-
ceptiori, is one and the same man during his whole life, in all and
each of his organs of sense. God is one and the same God, al
ways and everywhere. He is omnipresent, not virtually only,
but also substantially ; for virtue cannot subsist without sub
stance. In him are all things contained and moved ; yet neither
affects the other ; God suffers nothing from the motion of bodies ;
bodies find no resistance from the omnipresence of God. It is
allowed by all that the Supreme God exists necessarily ; and by
the same necessity he exists always and everywhere. Whence
also he is all similar, all eye, all ear, all brain, all arm, all powei

LIFE CF SIR ISAAC NEWTON. 37

to perceive, to understand, and to act ; but in a manner not at all
human, in a manner not at all corporeal, in a manner utterly un
known to us. As a blind man has no idea of colours, so have we
no idea of the manner by which the all-wise God perceives and
understands all things. He is utterly void of all body, and bodily
figure, and can therefore neither be seen, nor heard, nor touched ;
nor ought he to be worshipped under the representation of any
corporeal thing. We have ideas of his attributes, but what the
real substance of anything is we know not. In bodies we see
only their figures and colours, we hear only the sounds, we touch
only their outward surfaces, we smell only the smells, and taste
only the savours ; but their inward substances are not to be known,
either by our senses, or by any reflex act of our minds : much
less, then, have we any idea of the substance of God. We know
him only by his most wise and excellent contrivances of things,
and final causes ; we admire him for his perfections ; but we rev
erence and adore him on account of his dominion ; for we adore
him as his servants ; and a god without dominion, providence, and
final causes, is nothing else but Fate and Nature. Blind meta
physical necessity, which is certainly the same always and every
where, could produce no variety of things. All that diversity of
natural things which we find suited to different times and places
could arise from nothing but the ideas and will of a Being neces
sarily existing."

Thus, the diligent student of science, the earnest seeker of
truth, led, as through the courts of a sacred Temple, wherein, at
each step, new wonders meet the eye, till, as a crowning grace,
they stand before a Holy of Holies, and learn that all science and
all truth are one which hath its beginning and its end in the
knowledge of Him whose glory the heavens declare, and whose
handiwork the firmament showeth forth.

The introduction of the pure and lofty doctrines of the PRIN-
CIPIA was perseveringly resisted. Descartes, with his system of
vortices, had sown plausibly to the imagination, and error had
struck down deeply, and shot up luxuriantly, not only in the
popular, but in the scientific mind. Besides the idea — in itself so
simple and so grand — that the great masses of the planets were

38 LIFE OF SIR ISAAC NEWTON.

suspended in empty space, and retained in their orbits by an in
visible influence residing in the sun — was to the ignorant a thing
inconceivable, and to the learned a revival of the occult qualities
of the ancient physics. This remark applies particularly to the
continent. Leibnitz misapprehended ; Huygens in part rejected ;
John Bernouilli opposed ; and Fontenelle never received the doc
trines of the PRINCIPIA. So that, the saying of Voltaire is prob
ably true, that though Newton survived the publication of his
great work more than forty years, yet, at the time of his death,
lie had not above twenty followers out of England.

But in England, the reception of our author's philosophy was
rapid and triumphant. His own labours, while Lucasian Pro
fessor ; those of his successors in that Chair — Whiston and
Saunderson ; those of Dr. Samuel Clarke, Dr. Laughton, Roger
Cotes, and Dr. Bentley ; the experimental lectures of Dr. Keill
and Desaguliers ; the early and powerful exertions of David
Gregory at Edinburgh, and of his brother James Gregory at St.
Andrew's, tended to diffuse widely in England and Scotland a
knowledge of, and taste for the truths of the PRINCIPIA. Indeed,
its mathematical doctrines constituted, from the first, a regular
part of academical instruction ; while its physical truths, given to
the public in popular lectures, illustrated by experiments, had,
before the lapse of twenty ) ( ar.s, become familiar to, and adopted
by the general mind. Pemberton's popular " View of Sir Isaac
Newton's Philosophy" was published, in 1728 ; and the year after
ward, an English translation of the PRINCIPIA, and System of the
World, by Andrew Motte. And since that period, the labours of
Le Seur and Jacquier, of Thorpe, of Jebb, of Wright and others
have greatly contributed to display the most hidden treasures of
the PRINCIPIA.

About the time of the publication of the Principia, James II.,
bent on re-establishing the Romish Faith, had, among other ille
gal acts, ordered by mandamus, the University of Cambridge to
confer the degree of Master of Arts upon an ignorant monk.
Obedience to this mandate was resolutely refused. Newton was
one of the nine delegates chosen to defend the independence of
the University. They appeared before the High Court ; — and

LIFE OF SIR ISAAC NEWTON. 39

successfully : the king abandoned his design. The prominent
part which our author took in these proceedings, and his eminence
in the scientific world, induced his proposal as one of the parlia
mentary representatives of the University. He was elected, in
1688, and sat in the Convention Parliament till its dissolution.
After the first year, however, he seems to have given little or no
attention to his parliamentary duties, being seldom absent from
the University till his appointment in the Mint, in 1695.

Newton began his theological researches sometime previous to
1691 ; in the prime of his years, and in the matured vigour of
his intellectual powers. From his youth, as we have seen, he
had devoted himself with an activity the most unceasing, and an
energy almost superhuman to the discovery of physical truth ; —
giving to Philosophy a new foundation, and to Science a new
temple. To pass on, then, from the consideration of the material,
more directly to that of the spiritual, was a natural, nay, with so
large and devout a soul, a necessary advance. The Bible was to
him of inestimable worth. In the elastic freedom, which a pure
and unswerving faith in Him of Nazareth gives, his mighty facul
ties enjoyed the only completest scope for development. His
original endowment, however great, combined with a studious
application, however profound, would never, without this libera
tion from the dominion of passion and sense, have enabled him to
attain to that wondrous concentration and grasp of intellect, for
which Fame has as yet assigned him no equal. Gratefully he
owned, therefore, the'same Author in the Book of Nature and the
Book of Revelation. These were to him as drops of the same
unfathomable ocean ; — as outrayings of the same inner splendour ;
— as tones of the same ineffable voice ; — as segments of the same
infinite curve. "With great joy he had found himself enabled to
proclaim, as an interpreter, from the hieroglyphs of Creation, the
existence of a God : and now, with greater joy, and in the fulness
of his knowledge, and in the fulness of his strength, he laboured
to make clear, from the utterances of the inspired Word, the far
mightier confirmations of a Supreme Good, in all its glorious
amplitude of Being and of Attribute ; and to bring the infallible
workings thereof plainly home to the understandings and the

40 LIFE OF SIR ISAAC NEWTON.

•affections of his fellow-men ; and finally to add the weight of his
own testimony in favour of that Religion, whose truth is now. in
deed, " girded with the iron and the rock of a ponderous and co
lossal demonstration."

His work, entitled, OBSERVATIONS UPON THE PROPHECIES OF
HOLY WRIT, PARTICULARLY THE PROPHECIES OF DANIEL AND THE
APOCALYPSE OF ST. JOHN, first published in London, in 1733 4to.
consists of two parts : the one devoted to the Prophecies oi
Daniel, and the other to the Apocalypse of St. John. In the first
part, he treats concerning the compilers of the books of the Old
Testament ; — of the prophetic language ; — of the vision of the
four beasts ; — of the kingdoms represented by the feet of the
image composed of iron and clay ; — of the ten kingdoms repre
sented by the ten horns of the beast ; — of the eleventh horn of
Daniel's fourth beast ; of the power which should change times
and laws ;— of the kingdoms represented in Daniel by the ram
and he-goat ; — of the prophecy of the seventy weeks ; — of the
times of the birth and passion of Christ ; — of the prophecy of the
Scripture of Truth ; — of the king who doeth according to his will,
and magnified himself above every god, and honoured Mahuzzims,
and regarded not the desire of women ; — of the Mahuzzim, hon
oured by the king who doeth according to his will. In the sec
ond part, he treats of the time when the Apocalypse was written ,
of the scene of the vision, and the relation which the Apocalypse
has to the book of the law of Moses, and to the worship of God
in the temple ; — of the relation which the Apocalypse has to the
prophecies of Daniel, and of the subject of the prophecy itself
Newton regards the prophecies as given, not for the gratification
of man's curiosity, by enabling him to foreknow ; but for his con
viction that the world is governed by Providence, by witnessing
their fulfilment. Enough of prophecy, he thinks, has already
been fulfilled to afford the diligent seeker abundant evidence of
God's providence. The whole work is marked by profound
erudition, sagacity and argument.

And not less learning, penetration and masterly reasoning are
conspicuous in his HISTORICAL ACCOUNT OF Two NOTABLE
CORRUPTIONS OF SCRIPTURES IN A LETTER TO A FRIEND. This

LIFE OF SIR ISAAC NEWTON. 41

Treatise, first accurately published in Dr. Horsley's edition of his
works, relates to two texts : the one, 1 Epistle of St. John v. 7 ;
the other, 1 Epistle of St. Paul to Timothy iii. 16. As this
work had the effect to deprive the advocates of the doctrine of
the Trinity of two leading texts, Newton has been looked upon
as an Arian ; but there is absolutely nothing in his writings to
warrant such a conclusion.

His regaining theological works consist of the LEXICON PRO-
PHETICUM, which was left incomplete ; a Latin Dissertation on
the sacred cubit of the Jews, which was translated into English,
and published, in 1737. among the Miscellaneous Works of John
Greaves ; and FOUR LETTERS addressed to Dr. Bentlty, contain
ing some arguments in proof of a Deity. These Letters were
dated respectively : 10th December, 1692 ; 17th January, 1693 ;
25th February, 1693; and llth February, 1693— the fourth
bearing an earlier date than the third. The best faculties and
the profoundest acquirements of our author are convincingly
manifest in these lucid and powerful compositions. They were
published in 1756, and reviewed by Dr. Samuel Johnson.

Newton's religious writings are distinguished by their absolute
freedom from prejudice. Everywhere, throughout them, there
glows the genuine nobleness of soul. To his whole life, indeed,
we may here fitly extend the same observation. He was most
richly imbued with the very spirit of the Scriptures which he so
delighted to study and to meditate upon. His was a piety, so
fervent, so sincere and practical, that it rose up like a holy incense
from every thought and act. His a benevolence that not only
willed, but endeavoured the best for all. His a philanthropy
that held in the embracings of its love every brother-man.
His a toleration of the largest and the truest ; condemning per
secution in every, even its mildest form ; and kindly encouraging
each striving after excellence : — .1 toleration that came not of
indifference — for the immoral and the impious met with their
quick rebuke — but a toleration that came of the wise humbleness
and the Christian charity, which see, in the nothingness of self
and the almightiness of TRUTH, no praise for the ablest, and no
blame for th^ feeblest in their strugglings upward to light and life.

42 LIFE OF SIR ISAAC NEWTON,

Tn the winter of 1691-2, on returning from chapel, one morn
ing, Newton foima tnat a favourite little dog, called Diamond,
had overturned a lighted taper on his desk, and that several pa
pers containing the results of certain optical experiments, were
nearly consumed. His only exclamation, on perceiving his loss,
was, " Oh Diamond, Diamond, little knowest thou the mischiel
thou hast done," Dr. Brewster, in his life of our author, gives the
following extract from the manuscript Diary of Mr. Abraham De
La Pryme. a student in the University at the time of this oc
currence.

" 1692. February, 3. — What I heard to-day I must relate.
There is one Mr. Newton (whom I have very oft seen), Fellow
of Trinity College, that is mighty famous for his learning, being a
most excellent mathematician, philosopher, divine, &c. He has
been Fellow of the Royal Society these many years ; and among
other very learned books and tracts, he:s written one upon the mathe
matical principles of philosophy, which has given him a mighty
name, he having received, especially from Scotland, abundance of
congratulatory letters for the same ; but of all the books he ever
wrote, there was one of colours and light, established upon thou
sands of experiments which he had been twenty years of making,
and which had cost him many hundreds of pounds. This book
which he vaiued so much, and which was so much talked of, had
the ill luck to perish, and be utterly lost just when the learned
author was almost at pitting a conclusion at the same, after this
manner : In a winter's morning, leaving it among his other papers
on his study table while he went to chapel, the candle, which he
had unfortunately left burning there, too, catched hold by some
means of other papers, and they fired the aforesaid book, and ut
terly consumed it and several other valuable writings ; arid which
is most wonderful did no further mischief. But when Mr. New
ton came from chapel, and had seen what was done, every one
thought he would have run mad, he was so troubled thereat that
he was not himself for a month after. A long account of this his
system of colours you may find in the Transactions of the Royal
Society, which he had sent up to them long before this sad mis
chance happened unto him."

LIFE OF SIR ISAAC NEWTON. 43

It will be borne in mind that all of Newton's theological wri
tings, with the exception of the Letters to Dr. Bentley, were
composed before this event which, we must conclude, from
Pryme's words, produced a serious impression upon our author for
about a month. But M. Biot, in his Life of Newton, relying on a
memorandum contained in a small manuscript Journal of Huygens,
declares this occurrence to have caused a deran-gement of New
ton's intellect. M. Blot's opinions and deductions, however, as
well as those of La Place, upon this subject, were based upon
erroneous data, and have been overthrown by the clearest proof.
There is not, in fact, the least evidence that Newton's reason was,
for a single moment, dethroned ; on the contrary, the testimony
is conclusive that he was, at all times, perfectly capable of carry
ing on his mathematical, metaphysical and astronomical inquiries.
Loss of sleep, loss of appetite, and irritated nerves will disturb
somewhat the equanimity of the most serene ; and an act done, or
language employed, under such temporary discomposure, is not a
just criterion of the general tone and strength of a man's mind.
As to the accident itself, we may suppose, whatever might have
been its precise nature, that it greatly distressed him, and, still
further, that its shock may have originated the train of nervous
derangements, which afflicted him, more or less, for two years
afterward. Yet, during this very period of ill health, we find him
putting forth his highest powers. In 1692, he prepared for, and
transmitted to Dr. Wallis the first proposition of the Treatise on
Quadratures, with examples of it in first, second and third flux
ions. He investigated, in the same year, the subject of haloes ;
making and recording numerous and important observations rela
tive thereto. Those profound and beautiful Letters to Dr. Bentley
were written at the close of this and the beginning of the next
year. In October, 1693, Locke, who was then about publishing a
second edition of his work on the Human Understanding, request
ed Newton to reconsider his opinions on innate ideas. And in
1694, he was zealously occupied in perfecting his lunar theory ;
visiting Flamstead, at the Royal Observatory of Greenwich, in
September, and obtaining a series of lunar observations ; and

14 LIFE OF SIR ISAAC NEWTON.

commencing, in October, a correspondence with that distinguished
practical Astronomer, which continued till 1698.

We now arrive at the period when Newton permanently with
drew from the seclusion of a collegiate, and entered upon a more
active and public life. He was appointed Warden of the Mint,
in 1695, through the influence of Charles Montague, Chancellor
of the Exchequer, and afterward Earl of Halifax. The current
roin of the nation had been adulterated and debased, and Mon
tague undertook a re-coinage. Our author's mathematical and
chemical knowledge proved eminently useful in accomplishing
this difficult and most salutary reform. In 1699, he was pro
moted to the Mastership of the Mint — an office worth twelve or
fifteen hundred pounds per annum, and which he held during the
remainder of his life. He wrote, in this capacity, an official Re
port on the Coinage, which has been published ; he also prepared
a Table of Assays of Foreign Coins, which was printed at the
end of Dr. Arbuthnot's Tables of Ancient Coins, Weights, and
Measures, in 1727.

Newton retained his Professorship at Cambridge till 1703.
But he had, on receiving the appointment of Master of the Mint,
in 1699, made Mr. Whiston his deputy, with all the emoluments
of the office ; and, on finally resigning, procured his nomination to
the vacant Chair.

In January 1697, John Bernouilli proposed to the most distin
guished mathematicians of Europe two problems for solution.
Leibnitz, admiring the beauty of one of them, requested the time
for solving it to be extended to twelve months — twice the period
originally named. The delay was readily granted. Newton, how
ever, sent in, the day after he received the problems, a solution of
them to the President of the Royal Society. Bernouilli obtained
solutions from Newton, Leibinitz and the Marquis De L'Hopital ;
but Newton's though anonymous, he immediately recognised
" tanquam ungue leonem" as the lion is known by his claw.
We may mention here the famous problem of the trajectories
proposed by Leibnitz, in 1716, for the purpose of "feeling the
pulse of the English Analysts." Newton received the problem
about five o'clock in the afternoon, as he was returning from the

LIFE OF SIR ISAAC NEWTON. 45

Mint ; and though it was extremely difficult and he himself much
fatigued, yet he completed its solution, the same evening before
he went to bed.

The history of these problems affords, by direct comparison, a
striking illustration of Newton's vast superiority of mind. That
amazing concentration and grasp of intellect, of which we have
spoken, enabled him to master speedily, and, as it were, by a
single effort, those things, for the achievement of which, the many
would essay utterly in vain, and the very, very few attain only
after long and renewed striving. And yet, with a modesty as
unparalleled as his power, he attributed his successes, not to any
extraordinary sagacity, but solely to industry and patient thought.
Mr- kept the subject of consideration constantly before him, and
waited till the first dawning opened gradually into a full and
clear light ; never quitting, if possible, the mental process till the
object of it were wholly gained. He never allowed this habit of
meditation to appear in his intercourse with society ; but in the
privacy of his own chamber, or in the midst of his own family, he
gave himself up to the deepest abstraction. Occupied with some
interesting investigation, he would often sit down on his bedside,
after he rose, and remain there, for hours, partially dressed.
Meal-time would frequently come and pass unheeded ; so that,
unless urgently reminded, he would neglect to take the re
quisite quantity of nourishment. But notwithstanding his anx
iety to be left undisturbed, he would, when occasion required,
turn aside his thoughts, though bent upon the most intricate re
search, and then, when leisure served, again direct them to the
very point where they ceased to act : and this he seemed to ac
complish not so much by the force of his memory, as by the force
of his inventive faculty, before the vigorous intensity of which, no
subject, however abstruse, remained long unexplored.

Me was elected a member of the Royal Academy of Sciences
at Paris, in 1699, when that distinguished Body were empowered,
by a new charter, to admit a small number of foreign associates.
In 1700, he communicated to Dr. Halley a description of his re
flecting instrument for observing the moon's distance from the
fixed stars. This description was published in the Philosophical

46 LIFE OF SIR ISAAC NEWTON,

Transactions, in 1742. The instrument was the same as that
produced by Mr. Hadley, in 1731, and which, under the name of
Hadley's Quadrant, has been of so great use in navigation. On
the assembling of the new Parliament, in 1701, Newton was re-
elected one of the members for the University of Cambridge. In
1703, he was chosen President of the Royal Society of London,
to which office he was annually re-elected till the period of his
decease — about twenty-five years afterward.

Our author unquestionably devoted more labour to, and, in
many respects, took a greater pride in his Optical, than his other
discoveries. This science he had placed on a new and indestruc
tible basis ; and he wished not only to build, but to perfect the
costly and glowing structure. He had communicated, before the
publication of the PRINCIPIA, his most important researches on
light to the Royal Society, in detached papers which were inserted
in successive numbers of the Transactions ; but he did not pub
lish a connected view of these labours till 1704, when they appeared
under the title of OPTICS : OR, A TREATISE ON THE REFLEXIONS,
REFRACTIONS, INFLEXIONS AND COLOURS OF LIGHT. To this,
but to no subsequent edition, were added two Mathematical Trea
tises, entitled, TRACTATUS DUO DE SPECIEBUS ET MAGNITUDINE
FIGURARUM cuRViLiNEARUM ; the one bearing the title TRACTATUS
DE QUADRATURA CuRVARUM ; and the other, that of ENUMERATIO
LINEARUM TERTII ORDiNis. The publication of these Mathemati
cal Treatises was made necessary in consequence of plagiarisms
from the manuscripts of them loaned by the author to his friends.
Dr. Samuel Clarke published a Latin translation of the Optics, in
in 1706 ; whereupon he was presented by Newton, as a mark of
his grateful approbation, with five hundred pounds, or one hun
dred pounds for each of his children. The work was afterward
translated into French. It had a remarkably wide circulation,
and appeared, in several successive editions, both in England and
on the Continent. There is displayed, particularly on this Opti
cal Treatise, the author's talent for simplifying and communica
ting the profoundest speculations. It is a faculty rarely united to
that of the highest invention. Newton possessed both ; and thus
that mental perfectness which enabled him to create, to combine,

LIFE OF SIR ISAAC NEWTON. 47

and to teach, and so render himself, not the "ornament" cnly;
but inconceivably more, the pre-eminent benefactor of his species.

The honour of knighthood v/as conferred on our author in
1705. Soon afterward, he was a candidate again for the Repre
sentation of the University, but was defeated by a large majority.
It is thought that a more pliant man was preferred by both min
isters and electors. Newton was always remarkable for simplicity
of dress, and his only known departure from it was on this oc
casion, when he is said to have appeared in a suit of laced
clothes.

The Algebraical Lectures which he had, Juring nine years,
delivered at Cambridge, were published by Whiston, in 1707,
under the title of ARITHMETICS UNIVERSALIS, SINE DE COMPOSI
TIONS ET RESOLUTIONS ARITHMETICA LIBER. This publication
is said to have been a breach of confidence on Whiston's part. Mr.
Ralphson, not long afterward, translated the work into English ;
and a second edition of it, with improvements by the author, was
issued at London, 1712, by Dr. Machin. Subsequent editions,
both in English and Latin, with commentaries, have been published.

In June, 1709, Newton intrusted the superintendence of a
second edition of the PRINCIPIA to Roger Cotes, Plumian Pro
fessor of Astronomy at Cambridge. The first edition had been
sold off for some time. Copies of the work had become very
rare, and could only be obtained at several times their original
cost. A great number of letters passed oetween the author and
Mr. Cotes during the preparation of the edition, which finally
appeared in May, 1713. It had many alterations and improve
ments, and was accompanied by an admirable Preface from the
pen of Cotes.

Our author's early Treatise, entitled, ANALYSIS PER EQUATIONES
NUMERO TERMINORUM INFINITAS, as well as a small Tract, Gearing
the title of METHODUS DIFFERENTIALS, was published, witn nis
consent, in 1711. The' former of these, and the Treatise De
Quadratura Curvarum, translated into Englisn, witn a .arge com
mentary, appeared in 1745. His work, entitled. ARTIS ANA
LYTICS SPECIMINA, VEL GEOMETRIA ANALYTICA, was iirs; given
to the world in the edition of Dr. Horsley, 1779.

48 LIFE OF SIR ISAAC NEWTON.

It is a notable fact, in Newton's history, that he never volun*
tarily published any one of his purely mathematical writings
The cause of this unwillingness in some, and, in other instances,
of his indifference, or, at least, want of solicitude to put forth his
works may be confidently sought for in his repugnance to every
thing like contest or dispute. But, going deeper than this aver
sion, we find, underlying his whole character and running parallel
with all his discoveries, that extraordinary humility which always
preserved him in a position so relatively just to the behests of
time and eternity, that the infinite value of truth, and the utter
worthlessness of fame, were alike constantly present to him.
Judging of his course, however, in its more temporary aspect, as
bearing upon his immediate quiet, it seemed the most unfortunate.
For an early publication, especially in the case of his Method of
Fluxions, would have anticipated all rivalry, and secured him
from the contentious claims of Leibnitz. Still each one will solve
the problem of his existence in his own way, and, with a manlike
Newton, his own, as we conceive, could be no other than the best
way. The conduct of Leibnitz in this affair is quite irreconcilable
with the stature and strength of the man ; giant-like, and doing
nobly, in many ways, a giant's work, yet cringing himself into the
dimensions and performances of a common calumniator. Opening
in 1699, the discussion in question continued till the close of
Leibnitz's life, in 1716. We give the summary of the case as
contained in the Report of the Committee of the Royal Society,
the deliberately weighed opinion of which has been adopted as an
authoritative decision in all countries.

" We have consulted the letters and letter books in the custody
of the Royal Society, and those found among the papers of Mr.
John Collins, dated between the years 1669 and 1677, inclusive ;
and showed them to such as knew and avouched the hands of Mr.
Barrow, Mr. Collins, Mr. Oldenburg, and Mr. Leibnitz ; and
compared those of Mr. Gregory with one another, and with copies
of some of them taken in the hand of Mr. Collins ; and have
extracted from them what relates to the matter referred to us :
all which extracts, herewith delivered to you, we believe to be
genuine and authentic. And by these letters and papers wf
find: —

LIFE OF SIR ISAAC NEWTON. 49

" I. Mr. Leibnitz was in London in the beginning of the year
1673 ; and went thence in or about March, to Paris, where he
kept a correspondence with Mr. Collins, by means of Mr. Olden
burg, till about September, 1676, and then returned, by London
and Amsterdam, to Hanover: and that Mr. Collins was very free
in communicating to able mathematicians what he had received
from Mr, Newton and Mr. Gregory.

" II. That when Mr. Leibnitz was the first time in London,
he contended for the invention of another differential method,
properly so called ; and, notwithstanding he was shown by Dr.
Pell that it was Newton?s method, persisted in maintaining it to
be his own invention, by reason that he had found it by himself
without knowing what Newton had done before, and had much
improved it. And we find no mention of his having any other
differential method than Newton's before his letter of the 21st of
June, 1677, which was a year after a copy of Mr. Newton's letter
of the 10th of December, 1672, had been sent to Paris to be
communicated to him ; and above four years after Mr. Collins
began to communicate that letter to his correspondents ; in which
letter the method of fluxions was sufficiently described to any
intelligent person.

"III. That by Mr. Newton's letter, of the 13th of June, 1676
it appears that he had the method of fluxions above five years
before the writing of that letter. And by his Analysis per ^Equa-
tiones numero Terminorum Infmitas, communicated by Dr. Barrow
to Mr. Collins, in July, 1669, we find that he had invented the
method before that time.

"IV. That the differential method is one and the same with
the method of fluxions, excepting the name and mode of notation ;
Mr. Leibnitz calling those quantities differences which Mr. Newton
calls moments, or fluxions ; and marking them with a letter d — a
mark not used by Mr. Newton.

" And, therefore, we take the proper question to be, not who
invented this or that method, but, who was the first inventor of
the method ? And we believe that those who have reputed Mr.
Leibnitz the first inventor knew little or nothing of his correspond
ence with Mr. Collins and Mr. Oldenburg long before, nor of Mr.

50 LIFE OP SIR ISAAC NEWTON.

Newton's hiving that method above fifteen years before Mr
Leibnitz began to publish it in the Acta Eruditorum of Leipsic.

" For which reason we reckon Mr. Newton the first inventor ;
and are of opinion that Mr. Keill, in asserting the same, has been
no ways injurious to Mr. Leibnitz. And we submit to the judg
ment of the Society, whether the extract and papers, now pre
sented to you, together with what is extant, to the same pur
pose, in Dr. Wallis's third volume, may not deserve to be made
public."

This Report, with the collection of letters and manuscripts,
under the title of COMMERCIUM EPISTOLICUM D. JOHANNIS COLLINS

ET ALIORUM DE ANALYSI PROMOTA JuSSU SoCIETATIS REGIES

EDITUM, appeared accordingly in the early part of 1713. Its
publication seemed to infuse additional bitterness into the feelings
of Leibnitz, who descended to unfounded charges and empty
threats. He had been privy counsellor to the Elector of Han
over, before that prince was elevated to the British throne ; and
in his correspondence, in 1715 and 1716, with the Abbe Conti,
then at the court of George L, and with Caroline, Princess of
Wales, he attacked the doctrines of the PRINCIPIA, and indirectly
its author, in a manner very discreditable to himself, both as a
learned and as an honourable man. His assaults, however, were
triumphantly met; and, to the complete overthrow of his rival
pretensions, Newton was induced to give the finishing blow. The
verdict is universal and irreversible that the English preceded
the German philosopher, by at least ten years, in the invention
of fluxions. Newton could not have borrowed from Leibnitz ;
but Leibnitz might have borrowed from Newton. A new edition
of the Commercium Epistolicum was published in 1722-5 (?) ; but
neither in this, nor in the former edition, did our author take any
part. The disciples, enthusiastic, capable and ready, effectually
shielded, with the buckler of Truth, the character of the Master,
whose own conduct throughout was replete with delicacy, dignity
and justice. He kept aloof from the controversy — in which Dr.
Keill stood forth as the chief representative of the Newtonian
side — till the very last, when, for the satisfaction of the King,
George L. rather than for his own, he consented to put forth his

LIFE OF SI| L^.-vJ NEWTON. 5i

hand and firmly secure his rights upon a certain and impregnable,
basis.

A petition to have inventions for promoting the discovery of the
longitude at sea, suitably rewarded, was presented to the House
of Commons, in 1714. A committee, having been appointed to
investigate the subject, called upon Newton and others for their
opinions. That of our author was given in writing, A report,
favourable to the desired measure, was then taken up, and a bill
for its adoption subsequently passed.

On the ascension of George I., in 1714, Newton became an
object of profound interest at court. His position under govern
ment, his surpassing fame, his spotless character, and. above all,
his deep and consistent piety, attracted the reverent regard of the
Princess of Wales, afterward queen -consort to George II. She
was a woman of a highly cultivated mind, and derived the greatest
pleasure from conversing with Newton and corresponding with
Leibnitz. One day, in conversation with her, our author men
tioned and explained a new system of chronology, which he had
composed at Cambridge, where he had been in the habit " of
refreshing himself with history and chronology, when he wac
weary with other studies." Subsequently, in the year 1718, she
requested a copy of this interesting and ingenious work Newton,
accordingly, drew up an abstract of the system from the separate
papers in which it existed, and gave it to her on condition that it
should riot be communicated to any other person. Sometime
afterward she requested that the Abbe Conti might be allowed
to have a copy of it The author consented: and the abbe
received a copy of the manuscript, under the like injunction and
promise of secrecy. This manuscript bore the title of " A short
Chronicle, from the First Memory of Tilings in Europe, to the
Conquest of Persia, by Alexander the Great."

After Newton took up his residence in London, he lived in a
style suited to his elevated position and rank. He kept his car
riage, with an establishment of three male and three female serv
ants. But to everything like vain show and luxury he was utterly
averse. His household affairs, for the last twenty years of his
life, were under the charge of his niece, Mrs. Catherine Barton,

52 LIFE OF SIR ISAAC NEWTON.

wife and widow of Colonel Barton — a woman of great beauty and
accomplishment — and subsequently married to John Conduit, Esq.
At home Newton was distinguished by that dignified and gentle
hospitality which springs alone from true nobleness. On all pro
per occasions, he gave splendid entertainments, though without
ostentation. In society, whether of the palace or the cottage,
his manner was self-possessed and urbane ; his look benign and
affable ; his speech candid and modest ; his whole air undisturb
edly serene. He had none of what are usually called the singu
larities of genius ; suiting himself easily to every company —
except that of the vicious and wicked ; and speaking of himself
and others, naturally, so as never even to be suspected of vanity.
There was in him, if we may be allowed the expression, a WHOLE
NESS of nature, which did not admit of such imperfections and
weakness — the circle was too perfect, the law too constant, and
the disturbing forces too slight to suffer scarcely any of those
eccentricities which so interrupt and mar the movements of many
bright spirits, rendering their course through the world more like
that of the blazing meteor than that of the light and life-impart
ing sun. In brief, the words GREATNESS and GOODNESS could
not, humanly speaking, be more fitly employed than when applied
as the pre-eminent characteristics of this pure, meek and vene
rable sage.

In the eightieth year of his age, Newton was seized with
symptoms of stone in the bladder. His disease was pronounced
incurable. He succeeded, however, by means of a strict regimen,
and other precautions, in alleviating his complaint, and procuring
long intervals of ease. His diet, always frugai, was now extremely
temperate, consisting chiefly of broth, vegetables, and fruit, with,
now and then, a little butcher meat. He gave up the use of his
carriage, and employed, in its stead, when he went out, a chair.
All invitations to dinner were declined ; and only small parties
were received, occasionally, at his own house.

In 1724 he wrote to the Lord Provost of Edinburgh, offering
to contribute twenty pounds yearly toward the salary of Mr.
Maclaurin, provided he accepted the assistant Professorship of
Mathematics in the University of that place. Not only in the

LIFE OP SIR ISAAC NEWTON. 53

cause of ingenuity and learning, but in that of religion — in relieving
the poor and .assisting his relations, Newton annually expended
large sums. He was generous and charitable almost to a fault.
Those, he would often remark, who gave away nothing till they
died, never gave at all. His wealth had become considerable by
a prudent economy ; but he regarded money in no other light
than as one of the means wherewith he had been intrusted to do
good, and he faithfully employed it accordingly.

He experienced, in spite of all his precautionary measures, a
return of his complaint in the month of August, of the same year,
1 724, when he passed a stone the size of pea ; it came from him
in two pieces, the one at the distance of two day.s from the other.
Tolerable good health then followed for some months. In Janu
ary, 1725, however, he was taken with a violent cough and inflam
mation of the lungs. In consequence of this attack, he was pre
vailed upon to remove to Kensington, where his health greatly
improved. In February following, he was attacked in both feet
with the gout, of the approach of which he had received, a few
years before, a slight warning, and the presence of which now
produced a very beneficial change in his general health. Mr.
Conduit, his nephew, has recorded a curious conversation which
took place, at or near this time, between himself and Sir Isaac.

"I was, on Sunday night, the 7th March, 1724-5, at Kensing
ton, with Sir Isaac Newton, in his lodgings, just after he was out
of a fit of the gout, which he had had in both of his feet, for the
first time, in the eighty-third year of his age. He was better after
it, and his head clearer and memory stronger than I had known
them for some time. He then repeated to me, by way of dis
course, very distinctly, though rather in answer to my queries,
than in one continued narration, what he had often hinted to me
before, viz. : that it was his conjecture (he would affirm nothing)
that there was a sort of revolution in the heavenly bodies ; that
the vapours and light, emitted by the sun, which had their sedi
ment, as water and other matter, had gathered themselves, by
degrees, into a body, and attracted more matter from the planets,
and at last made a secondary planet (viz. : one of those that go
round another planet), and then, by gathering to them, and

54 LIFE OF SIR ISAAC NEWTON.

attracting more matter, became a primary planet ; and then, bf
increasing still, became a comet, which, after certain revolutions,
by coming nearer and nearer to the sun, had all its volatile parts
condensed, and became a matter tit to recruit and replenish the
sun (which must waste by the constant heat and light it emitted),
as a faggot would this fire if put into it (we were sitting by a
wood fire), and that that would probably be the effect of the
comet of 1680, sooner or later ; for, by the observations made
upon it, it appeared, before it came near the sun, with a tail only
two or three degrees long ; but, by the heat it contracted, in going
so near the sun, it seemed to have a tail of thirty or forty degrees
when it went frpm it ; that he could not say when this comet
would drop into the sun ; it might perhaps have five or six revo
lutions more first, but whenever it did it would so much increase
the heat of the sun that this earth would be burned, and no ani
mals in it could live. That he took the three phenomena, seen
by Hipparchus, Tycho Brahe, and Kepler's disciples, to have been
of this kind, for he could not otherwise account for an extraor
dinary light, as those were, appearing, all at once, among the
the fixed stars (all which he took to be suns, enlightening other
planets, as our sun does ours), as big as Mercury or Venus seems
to us, and gradually diminishing, for sixteen months, and then
sinking into nothing. He seemed to doubt whether there were
not intelligent beings, superior to us, who superintended these
revolutions of the heavenly bodies, by the direction of the Supreme
Being. He appeared also to be very clearly of opinion that the
inhabitants of this world were of short date, and alledged, as one
reason for that opinion, that all arts, as letters, ships, printing,
needle, &c., were discovered within the memory of history, which
could not have happened if the world had been eternal ; and that
there were visible marks of ruin upon it which could not be
effected by flood only. When I asked him how this earth could
have been repeopled if ever it had undergone the same fate
it was threatened with hereafter, by the comet of 1680, he
answered, that required the power of a Creator. He said he
took all the planets to be composed of the same matter with this
earth, viz. : earth, water, stones, &c.3 but variously concocted. J

LIFE OP SIR ISAAC NEWTON. 55

asked him why he would not publish his conjectures, as conjec
tures, and instanced that Kepler had communicated his ; and
though he had not gone near so far as Kepler, yet Kepler's
guesses were so just and happy that they had been proved and
demonstrated by him. His answer was, " I do not deal in con
jectures." But, on my talking to him about the four observations
that had been made of the comet of 1680, at 574 years' distance,
and asking him the particular times, he opened his Principia,
which laid on the table, and showed me the particular periods,
viz.: 1st. The Julium Sidus, in the time of Justinian, in 1106,
in 1680.

" And I, observing that he said there of that comet, ' incidet
in corpus solis,' and in the next paragraph adds, ' stellae fixae
refici possunt,' told him I thought he owned there what we had
been talking about, viz. : that the comet would drop into the sun,
and that fixed stars were recruited and replenished by comets
when they dropped into them ; and, consequently, that the sun
would be recruited too ; and asked him why he would not own as
fully what he thought of the sun as well as what he thought of
the fixed stars. He said, 'that concerned us more;' and, laugh
ing, added, that he had said enough for people to know his
meaning."

In the summer of 1725, a French translation of the chronolo
gical MS., of which the Abbe Conti had been permitted, some
time previous, to have a copy, was published at Paris, in violation
of all good faith. The Punic Abbe had continued true to his
promise of secrecy while he remained in England ; but no sooner
did he reach Paris than he placed the manuscript into the hands
of M. Freret, a learned antiquarian, who translated the work, and
accompanied it with an attempted refutation of the leading points
of the system. In November, of the same year, Newton received
a presentation copy of this publication, which bore the title of
ABREGE DE CHRONOLOGIE DE M. LE CHEVALIER NEWTON, FAIT

PAR LUI-MEME, ET TRADUIT SUR LE MANUSCRIPT ANGLAIS. Soon

afterward a paper entitled, REMARKS ON TFE OBERVATIONS MADE
ON A CHRONOLOGICAL INDEX OF SIR ISAAC NE.WTON, TRANSLATED
INTO FRENCH BY THE OBSERVATOR, ANL PUBLISHED AT PARIS,

56 LIFE OF SIR ISAAC NEWTON,

was drawn up by our author, and printed in the Philosophical
Transactions for 1725. It contained a history of the whole
matter, and a triumphant reply to the objections of M. Freret.
This answer called into the field a fresh antagonist, Father Soueiet,
whose five dissertations on this subject were chiefly remarkable
for the want of knowledge and want of decorum, which they
displayed. In consequence of these discussions, Newton was in
duced to prepare his larger work for the press, and had nearly
completed it at the time of his death. It was published in 1728,
under the title of THE CHRONOLOGY OF THE ANCIENT KINGDOMS
AMENDED, TO WHICH is PREFIXED A SHORT CHRONICLE FROM THE

FIRST MEMORY OF THINGS IN EUROPE TO THE CONQUEST OF

PERSIA BY ALEXANDER THE GREAT. It consists of six chap
ters: 1. On the Chronology of the Greeks; according to Whis-
ton, our author wrote out eighteen copies of this chapter with his
own hand, differing little from one another. 2. Of the Empire
of Egypt; 3. Of the Assyrian Empire; 4. Of the two contempo
rary Empires of the Babylonians and Medes ; 5. A Description
of the Temple of Solomon ; 6. Of the Empire of the Persians ;
this chapter was not found copied with the other five, but as it
was discovered among his papers, arid appeared to be a continu
ation of the same work, the Editor thought proper to add it
thereto. Newton's LETTER TO A PERSON OF DISTINCTION WHO

HAD DESIRED HIS OPINION OF THE LEARNED BlSHO^ LLOYD'S

HYPOTHESIS CONCERNING THE FORM OF THE MOST ANCIENT
^EAR, closes this enumeration of his Chronological Writings.

A ihird edition of the PRINCIPIA appeared in 1726, with many
changes and additions. About four years were consumed in its
preparation and publication, which were under the superintend-
ance of Dr. Henry Pemberton, an accomplished mathematician,
and the author of "A VIEW OF SIR ISAAC NEWTON'S PHILO
SOPHY." 1728. This gentleman enjoyed numerous opportunities
of conversing with the aged and illustrious author. " I found,"
says Pemberton, " he had read fewer of the modern mathemati
cians than one could have expected; but his own prodigious
invention readily supplied him with what he might have an occa
sion for in the pursuit of any subject he undertook. I have often

LIFE OF SIR ISAAC NEWTON. 57

heard him censure the handling geometrical subjects ly algebraic
calculations ; and his book of Algebra he called by the name of
Universal Arithmetic, in opposition to the injudicious title of
Geometry, which Descartes had given to the treatise, wherein he
shows how the geometer may assist his invention by such kind
of computations. He thought Huygens the most elegant of any
mathematical writer of modern times, and the most just imitator
of the ancients. Of their taste and form of demonstration, Sir
Isaac always professed himself a great admirer. I have heard
him even censure himself for not following them yet more closely
than he did ; and speak with regret of his mistake at the begin
ning of his mathematical studies, in applying himself to the works
of Descartes and other algebraic writers, before he had considered
the elements of Euclid with that attention which so excellent a
writer deserves."

" Though his memory was much decayed," continues Dr. Pem-
berton, "he perfectly understood his own writings." And even
this failure of memory, we would suggest, might have been more
apparent than real, or, in medical terms, more the result of func
tional weakness than organic decay. Newton seems never to
have confided largely to his memory : and as this faculty mani
fests the most susceptibility to cultivation ; so, in the neglect of
due exercise, it more readily and plainly shows a diminution of
its powers.

Equanimity and temperance had, indeed, preserved Newton
singularly free from all mental and bodily ailment. His hair was,
to the last, quite thick, though as white as silver. He never
made use of spectacles, and lost but one tooth to the day of his
death. He was of middle stature, well-knit, and, in the latter
part of his life, somewhat inclined to be corpulent. Mr. Conduit
says, " he had a very lively and piercing eye, a comely and gra
cious aspect, with a fine head of hair, white as silver, without any
baldness, and when his peruke was off was a venerable sight."
According to Bishop Atterbury, "in the whole air of his face and
make there was nothing of that penetrating sagacity which
appears in his compositions. He had something rather languid
in his look and manner which did not raise any great expectation

58 LIFE OF SIR ISAAC NEWTON.

in those who did not know him." Hearne remarks, " Sir Isaac
was a man of no very promising aspect. He was a short, well-
set man. He was full of thought, and spoke very little in com
pany, so that his conversation was not agreeable. When he rode
in his coach, one arm would be out of his coach on one side and
the other on the other." These different accounts we deem
easily reconcilable. In the rooms of the Royal Society, in the
street, or in mixed assemblages, Newton's demeanour — always
courteous, unassuming and kindly — still had in it the overawings
of a profound repose and reticency, out of which the communica
tive spirit, and the "lively and piercing eye" would only gleam
in the quiet and unrestrained freedom of his own fire-side.

" But this I immediately discovered in him," adds Pemberton,
still further, "which at once both surprised and charmed me.
Neither his extreme great age, nor his universal reputation had
rendered him stiff in opinion, or in any degree elated. Of this I
had occasion to have almost daily experience. The remarks I
continually sent him by letters on his Principia, were received
with the utmost goodness. These were so far from being any
ways displeasing to him, that, on the contrary, it occasioned him
to speak many kind things of me to my friends, and to honour me
with a public testimony of his good opinion." A modesty, open
ness, and generosity, peculiar to the noble and comprehensive
spirit of Newton. " Full of wisdom and perfect in beauty," yet
not lifted up by pride nor corrupted by ambition. None, how
ever, knew so well as himself the stupendousness of his discoveries
in comparison with all that had been previously achieved ; and
none realized so thoroughly as himself the littleness thereof in
comparison with the vast region still unexplored. A short time
before his death he uttered this memorable sentiment: — " I do not
know what I may appear to the world ; but to myself I seem to
have been only like a boy playing on the sea-shore, and diverting
myself in now and then finding a smoother pebble or a prettier
shell than ordinary, while the great ocean of truth lay all undis
covered before me." How few ever reach the shore even, much
less find "a smoother pebble or a prettier shell!"

Newton had now resided about two years at Kensington ; and

LIFE OF SIR ISAAC NEWTON. 59

the air which he enjoyed there, and the state of absolute rest,
proved of great benefit to him. Nevertheless he would occasion
ally go to town. And on Tuesday, the 28th of February, 1727,
he proceeded to London, for the purpose of presiding at a meeting
of the Royal Society. At this time his health was considered,
by Mr. Conduit, better than it had been for many years. But
the unusual fatigue he was obliged to suffer, in attending the
meeting, and in paying and receiving visits, speedily produced a
violent return of the affection in the bladder. He returned to
Kensington on Saturday, the 4th of March. Dr. Mead and Dr.
Cheselden attended him ; they pronounced his disease to be the
stone, and held out no hopes of recovery. On Wednesday, the
15th of March, he seemed a little better; and slight, though
groundless, encouragement was felt that he might survive the
attack. From the very first of it, his sufferings had been intense.
Paroxysm followed paroxysm, in quick succession : large drops
)f sweat rolled down his face ; but not a groan, not a complaint,
not the least mark of peevishness or impatience escaped him :
and during the short intervals of relief, he even smiled and con
versed with his usual composure and cheerfulness. The flesh
quivered, but the heart quaked not ; the impenetrable gloom was
settling down : the Destroyer near ; the portals of the tomb
opening, still, arnid this utter wreck and dissolution of the mortal,
the immortal remained serene, unconquerable : the radiant light
broke through the gathering darkness ; and Death yielded up its
sting, and the grave its victory. On Saturday morning, 18th,
he read the newspapers, and carried on a pretty long conversation
with Dr. Mead. His senses and faculties were then strong and
vigorous ; but at six o'clock, the same evening, he became insen
sible ; and in this state he continued during the whole of Sunday,
and till Monday, the 20th, when he expired, between one and
two o'clock in the morning, in the eighty-fifth year of his age.

And these were the last days of Isaac Newton. Thus closed
the career of one of earth's greatest and best men. His mission
was fulfilled. Unto the Giver, in many-fold addition, the talents
were returned. While it was yet day he had worked ; and for
the night that quickly cometh he was not unprepared. Full of

60 LIFE OF SIR ISAAC NEWTON.

years, ind full of honours, the heaven-sent was recalled ; and, in
the confidence of a " certain hope," peacefully he passed awa}'
into the silent depths of Eternity.

His body was placed in Westminster Abbey, with the state
and ceremonial that usually attended the interment of the most
distinguished. In 1731, his relatives, the inheritors of his personal
estate, erected a monument to his memory in the most conspicu
ous part of the Abbey, which had often been refused by the dean
and chapter to the greatest of England's nobility. During the
same year a medal was struck at the Tower in his honour ; arid,
in 1755, a full-length statue of him, in white marble, admirably
executed, by Roubiliac, at the expense of Dr. Robert Smith, was
erected in the ante-chamber of Trinity College, Cambridge.
There is a painting executed in the glass of one of the windows
of the same college, made pursuant to the will of Dr. Smith, who
left five hundred pounds for that purpose.

Newton left a personal estate of about thirty-two thousand
pounds. It was divided among his four nephews and four nieces
of the half blood, the grand-children of his mother, by the Reve
rend Mr. Smith. The family estates of Woolsthorpe arid Sustern
fell to John Newton, the heir-at-law, whose great grand-father
was Sir Isaac's uncle. Before his death he made an equitable
distribution of his two other estates : the one in Berkshire to the
sons and daughter of a brother of Mrs. Conduit ; and the other,
at Kensington, to Catharine, the only daughter of Mr. Conduit,
and who afterward became Viscountess Lymington. Mr. Con
duit succeeded to the offices of the Mint, the duties of which he
had discharged during the last two years of Sir Isaac's life.

Our author's works are found in the collection of Castilion,
Berlin, 1744, 4to. 8 torn.; in Bishop Horsley's Edition, London,
1779, 4to. 5 vol.; in the Biographia Brittannica, &c. Newton
also published Bern. Varcnii Geographia, &c., 1681, 8vo.
There are, however, numerous manuscripts, letters, and other
papers, which have never been given to the world: these are
preserved, in various collections, namely, in the library of Trinity
College, Cambridge ; in the library of Corpus Christi College,
Oxford ; in the library of Lord Macclesfield : and, lastly arid

LIFE OF SIR ISAAC NEWTON. 61

chiefly, in the possession of the family of the Earl of Portsmouth,
through the Viscountess Lymington.

Everything appertaining to Newton has been kept and che
rished with peculiar veneration. Different memorials of him are
preserved in Trinity College, Cambridge ; in the rooms of the
Royal Society, of London : and in the Museum of the Royal
Society of Edinburgh.

The manor-house, at Woolsthorpe, was visited by Dr. Stuke
ley, in October, 1721, who, in a letter to Dr. Mead, written in
1727, gave the following description of it: — " 'Tis built of stone,
as is the way of the country hereabouts, and a reasonably good
one. They led me up stairs and showed me Sir Isaac's stud}-,
where I supposed he studied, when in the country, in his younger
days, or perhaps when he visited his mother from the University.
I observed the shelves were of his own making, being pieces of
deal boxes, which probably he sent his books and clothes down
in on those occasions. There were, some years ago, two or threr
hundred books in it of his father-in-law, Mr. Smith, which Sir
Isaac gave to Dr. Newton, of our town." The celebrated apple-
tree, the fall of one of the apples of which is said to have turned
the attention of Newton to the subject of gravity, was destroyed
by the wind about twenty years ago ; but it has been preserved
in the form of a chair. The house itself has been protected with
religious care. It was repaired in 1798, and a tablet of white
marble put up in the room where our author was born, with the
follow, ng inscription : —

" Sir Isaac Newton, son of John Newton, Lord of the Manor
of Woolsthorpe, was born in this room, on the 25th of December,
1642."

Nature and Nature's Laws wei-e hid in night,
God said, " Let NEWTON be," and all was light.

THE PEINCIPIA.

THE AUTHOR'S PREFACE

SINCE the ancients (as we are told by Pappus), made great account oi
the science of mechanics in the investigation of natural things : and the
moderns, laying aside substantial forms and occult qualities, have endeav
oured to subject the phenomena of nature to the laws of mathematics, I
have in this treatise cultivated mathematics so far as it regards philosophy.
The ancients considered mechanics in a twofold respect ; as rational, which
proceeds accurately by demonstration ; and practical. To practical me
chanics all the manual arts belong, from which mechanics took its name.
Rut as artificers do not work with perfect accuracy, it comes to pass that
mechanics is so distinguished from geometry, that what is perfectly accu
rate is called geometrical , what is less so, is called mechanical. But the
errors are not in the art, but in the artificers. He that works with less
accuracy is an imperfect mechanic ; and if any could work with perfect
accuracy, he would be the most perfect mechanic of all ; for the description
if right lines and circles, upon which geometry is founded, belongs to me
chanics. Geometry does not teach us to draw these lines, but requires
them to be drawn ; for it requires that the learner should f.rst be taught
to describe these accurately, before he enters upon geometry ; then it shows
how by these operations problems may be solved. To describe right lines
and circles are problems, but not geometrical problems. The solution of
these problems is required from mechanics ; and by geometry the use of
them, when so solved, is shown ; and it is the glory of geometry that from
those few principles, brought from without, it is able to produce so many
things. Therefore geometry is founded in mechanical practice, and is
nothing but that part of universal mechanics which accurately proposes
and demonstrates the art of measuring. But since the manual arts are
chiefly conversant in the moving of bodies, it comes to pass that geometry
is commonly referred to their magnitudes, and mechanics to their motion.
In this sense rational mechanics will be the science of motions resulting
from any forces whatsoever, and of the forces required to produce any mo
tions, accurately proposed and demonstrated. This part of mechanics was

i:;vm THE AUTHOR & PREFACE.

cultivated by the ancients in the five powers which relate to manual arts,
who considered gravity (it not being a manual power), ho Otherwise than
as it moved weights by those powers. Our design not respecting arts, but
philosophy, and our subject not manual but natural powers, we consider
chiefly those things which relate to gravity, levity, elastic force, the resist
ance of fluids, and the like forces, whether attractive or impulsive ; and
therefore we offer this work as the mathematical principles :f philosophy ; for
all the difficulty of philosophy seems to consist in this — from the phenom
ena of motions to investigate the forces of nature, and then from these
forces to demonstrate the other phenomena ; and to this end the general
propositions in the first and second book are directed. In the third book
we give an example of this in the explication of the System of the World :
for by the propositions mathematically demonstrated in the former books,
we in the third derive from the celestial phenomena the forces of gravity
with which bodies tend to the sun and the several planets. Then from these
forces, by other propositions which are also mathematical, we deduce the mo
tions of the planets, the comets, the moon, and the sea. I wish we could do-
rive the rest of the phenomena of nature by the same kind of reasoning from
mechanical principles; for I am induced by many reasons to suspect that
they may all depend upon certain forces by which the particles of bodies.
by some causes hitherto unknown, are either mutually impelled towards
each other, and cohere in regular figures, or are repelled and recede from
each other; which forces being unknown, philosophers have hitherto at
tempted the search of nature in vain ; but I hope the principles here laid
down will afford some light either to this or some truer method of philosophy.
In the publication of this work the most acute and universally learned
Mr. Edmund H alley not only assisted me with his pains in correcting the
press and taking care of the schemes, but it was to his solicitations that its
becoming public is owing ; for when he had obtained of me my demonstra
tions of the figure of the celestial orbits, he continually pressed me to com
municate the same to the Royal Societ //, who afterwards, by their kind en
couragement and entreaties, engaged me to think of publishing them. But
after I had begun to consider the inequalities of the lunar motions, and
had entered upon some other things relating to the laws and measures oi
gravity, and other forces : and the figures that would be described by bodies
attracted according to given laws ; and the motion of several bodies moving
among themselves; the motion of bodies in resisting mediums; the forces,
densities, and motions, of rn( Hums ; the orbits of the comets, and such like ;

Ixix

deferred that publication till I had made a searcli into those matters, and
could put forth the whole together. What relates to the lunar motions (be
ing imperfect), I have put all together in the corollaries of Prop. 66, to
avoid being obliged to propose and distinctly demonstrate the several things
there contained in a method more prolix than the subject deserved, and in
terrupt the series of the several propositions. Some things, found out after
the rest, I chose to insert in places less suitable, rather than change the
number of the propositions and the citations. I heartily beg that what 1
have here done may be read with candour; and that the defects in a
subject so difficult be not so much reprehended as kindly supplied, and in
vestigated by new endeavours of mv readers.

ISAAC NEWTON.

Cambridge, Trinity Coupge May 8, l«iHB.

In the second edition the second section of the first book was enlarged.
In the seventh section of the second book the theory of the resistances of fluids
was more accurately investigated, and confirmed by new experiments. In
the third book the moon's theory and the profession of the equinoxes were
more fully deduced from their principles ; and the theory of the comets
was confirmed by more examples of the calculati >n of their orbits, done
also with greater accuracy.

In this third edition the resistance of mediums is somewhat more largely
handled than before; and new experiments of the resistance of heavy
bodies falling in air are added. In the third book, the argument to prove
that the moon is retained in its orbit by the force of gravity is enlarged
on ; and there are added new observations of Mr. Pound's of the proportion
of the diameters of Ju.piter to each other : there are, besides, added Mr.
Kirk's observations of the comet in 16SO ; the orbit of that comet com
puted in an ellipsis by Dr. Halley ; and the ortit of the comet in
computed by Mr. Bradley,

OOK I.

THE

MATHEMATICAL PRINCIPLES

OF

NATURAL PHILOSOPHY

DEFINITIONS.

DEFINITION I.
77w? quantity of matter is the measure of the same, arising from its

density and hulk conjutictly.

THUS air of a double density, in a double space, is quadruple in quan-
ti ty ; in a triple space, sextuple in quantity. The same thing is to be un
derstood of snow, and fine dust or powders, that are condensed by compres
sion or liquefaction • and of all bodies that are by any causes whatever
differently condensed. I have no regard in this place to a medium, if any
such there is, that freely pervades the interstices between the parts oi
bodies. It is this quantity that I mean hereafter everywhere under the
name of body or mass. And the same is known by the weight of each
body ; for it is proportional to the weight, as I have found by experiments
on pendulums, very accurately made, which shall be shewn hereafter.

DEFINITION II.

The quantity of motion is the measure nf tlie same, arising from the

velocity and quantity of matter corjunctly.

The motion of the whole i<! the sum of the motions of all the parts ; and
therefore in a body double in quantity, with equal velocity, the motion is
iouble ; with twice the velocity, it is quadruple,

DEFINITION III.

The vis insita, or innate force of matter, is a power of resisting, hy
which every body, as much as in it lies, endeavours to persevere in its
present stale, whether it be of rest, or of moving uniformly forward
in a right line.
This force is ever proportional to the body whose force it is ; and differs

nothing from the inactivity of the mass, but in our manner of conceiving

T4 THE MATHEMATICAL PRINCIPLES

it. A body, from the inactivity of matter, is not without difficulty put out
of its state of rest or motion. Upon which account, this vis insita, may,
by a most significant name, be called vis inertia, or force of inactivity.
Hut a body exerts this force only, when another force, impressed upon it,
endeavours to change its condition ; and the exercise of this force may bo
considered both as resistance and impulse ; it is resistance, in so far as the
body, for maintaining its present state, withstands the force impressed; it
is impulse, in so far as the body, by not easily giving way to the impressed
force of another, endeavours to change the state of that other. Resistance
is usually ascribed to bodies at rest, and impulse to those in motion;
but motion and rest, as commonly conceived, are only relatively distin
guished ; nor are those bodies always truly at rest, which commonly are
taken to be so.

DKFLMTIOX IV.

Ait impressed force is an action exerted upon a body, in order to change
its state, either of rest, or of moving uniformly forward in a right
line.

This force consists in the action only; and remains no longer in the
body, when the action is over. For a body maintains every new state it
acquires, by its vis inertice only. Impressed forces are of differe.it origins •
as from percussion, from pressure, from centripetal force.

DEFINITION V.

A centripetal force is that by which bodies are drawn or impelled, or any

way tend, towards a point as to a centre.

Of this sort is gravity, by which bodies tend to the centre of the earth
magnetism, by which iron tends to the loadstone ; and that force, what
ever it is, by which the planets are perpetually drawn aside from the rec
tilinear motions, which otherwise they would pursue, and made to revolve
in curvilinear orbits. A stone, whirled about in a sling, endeavours to re
cede from the hand that turns it ; and by that endeavour, distends the
sling, and that with so much the greater force, as it is revolved with the
greater velocity, and as soon as ever it is let go, flies away. That force
which opposes itself to this endeavour, and by which the sling perpetually
draws back the stone towards the hand, and retains it in its orbit, because
it is directed to the hand as the centre of the orbit, I call the centripetal
force. And the same thing is to be understood of all bodies, revolved in
any orbits. They all endeavour to recede from the centres of their orbits ;
and wore it not for the opposition of a contrary force which restrains them
to, and detains them in their orbits, which I therefore call centripetal, would
tiy off in right lines, with an uniform motion. A projectile, if it was not
for the force of gravity, would not deviate towards the earth, tut would

OF NATUJIAL PHILOSOPHY. 7fl

go off from it in a right line, and that with an uniform motion,, if the re
sistance of the air was taken away. It is by its gravity that it is drawn
aside perpetually from its rectilinear course, and made to deviate towards
the earth, more or less, according to the force of its gravity, and the velo
city of its motion. The less its gravity is, for the quantity of its matter,
or the greater the velocity with which it is projected, the less will it devi
ate from a rectilinear course, and the farther it will go. If a leaden balJ,
projected from the top of a mountain by the force of gunpowder with a
given velocity, and in a direction parallel to the horizon, is carried in a
curve line to the distance of two miles before it falls to the ground ; the
same, if the resistance of the air were taken away, with a double or decuple
velocity, would fly twice or ten times as far. And by increasing the velo
city, we may at pleasure increase the distance to which it might be pro
jected, and diminish the curvature of the line, which it might describe, till
at last it should fall at the distance of 10, 30, or 90 degrees, or even might
go quite round the whole earth before it falls ; or lastly, so that it might
never fall to the earth, but go forward into the celestial spaces, and pro
ceed in its motion in iiifiuitum. And after the same manner that a pro
jectile, by the force of gravity, may be made to revolve in an orbit, and go
round the whole earth, the moon also, either by the force of gravity, if it
is endued with gravity, or by any other force, that impels it towards the
earth, may be perpetually drawn aside towards the earth, out of the r&ti-
linear way, which by its innate force it would pursue; and would be made
to revolve in the orbit which it now describes ; nor could the moon with
out some such force, be retained in its orbit. If this force was too small,
it would not sufficiently turn the moon out of a rectilinear course : if it
was too great, it would turn it too much, arid draw down the moon from
its orbit towards the earth. It is necessary, that the force be of a just
quantity, and it belongs to the mathematicians to find the force, that may
serve exactly to retain a body in a given orbit, with a given velocity ; and
vice versa, to determine the curvilinear way, into which a body projected
from a given place, with a given velocity, may be made to deviate from
its natural rectilinear way, by means of a given force.

The quantity of any centripetal force may be considered as of three
kinds; aboolu'e, accelerative, and motive.

DEFINITION VI.

The absolute quantity of a centripetal force is the measure f>f the same
proportional to the efficacy of the cause that propagates it from the cen
tre, through the spaces round about.
Thus the magnetic force is greater in one load-stone and less in another

according to their sizes and strength of intensity.

76 THE MATHEMATICAL PRINCIPLES

DEFINITION VII.

The accelerative quantity of a centripetal force is the measure, of tht
same, proportional to the velocity which it generates in a given time.

Thus the force of the same load-stone is greater at a less distance, and
less at a greater : also the force of gravity is greater in valleys, less on
tops of exceeding high mountains ; and yet less (as shall hereafter be shown),
at greater distances from the body of the earth ; but at equal distan
ces, it is the same everywhere ; because (taking away, or allowing for, the
resistance of the air), it equally accelerates all falling bodies, whether heavy
or light, great or small.

DEFINITION VIII.

TJie motive quantity of a centripetal force, is the measure of the samt\

proportional to the motion which it generates in a given twip.

Thus the weight is greater in a greater body, less in a less body ; and.
in the same body, it is greater near to the earth, and less at remoter dis
tances. This sort of quantity is the centripetency, or propension of the
whole body towards the centre, or, as I may say, its weight ; and it is al
ways known by the quantity of an equal and contrary force just sufficient
to Ifinder the descent of the body.

These quantities of forces, we may, for brevity's sake, call by the names
of motive, accelerative, and absolute forces ; and, for distinction's sake, con
sider them, with respect to the bodies that tend to the centre ; to the places
of those bodies ; and to the centre of force towards which they tend ; that
is to say, I refer the motive force to the body as an endeavour and propen
sity of the whole towards a centre, arising from the propensities of the
several parts taken together ; the accelerative force to the place of the
body, as a certain power or energy diffused from the centre to all places
around to move the bodies that are in them : and the absolute force to
the centre, as endued with some cause, without which those motive forces
would not be propagated through the spaces round about ; whether that
cause be some central body (siuh as is the load-stone, in the centre of the
magnetic force, or the earth in the centre of the gravitating force), or
anything else that does not yet appear. For I here design only to give a
mathematical notion of those forces, without considering their physical
causes and seats.

Wherefore the accelerative force will stand in the same relation to the
motive, as celerity does to motion. For the quantity of motion arises from
the celerity drawn into the quantity of matter : and the motive force arises
from the accelerative force drawn into the same quantity of matter. For
the sum of the actions of the accelerative force, upon the several ; articles
of the body, is the motive force of the whole. Hence it is, that near the

OF NATURAL PHILOSOPHY. 77

surface of the earth, where the accelerative gravity, or force productive of
gravity, in all bodies is the same, the motive gravity or the weight is as
the body : but if we should ascend to higher regions, where the accelerative
gravity is less, the weight would be equally diminished, and would always
be as the product of the body, by the accelerative gravity. So in those re
gions, where the accelerative gravity is diminished into one half, the weight
of a body two or three times less, will be four or six times less.

I likewise call attractions and impulses, in the same sense, accelerative,
and motive ; and use the words attraction, impulse or propensity of any
sort towards a centre, promiscuously, and indifferently, one for another ;
considering those forces not physically, but mathematically : wherefore, the
reader is not to imagine, that by those words, I anywhere take upon me to
define the kind, or the manner of any action, the causes or the physical
reason thereof, or that I attribute forces, in a true and physical sense, to
certain centres (which are only mathematical points) ; when at any time I
happen to speak of centres as attracting, or as endued with attractive
powers.

SCHOLIUM.

Hitherto I have laid down the definitions of such words as are less
known, and explained the sense in which I would have them to be under
stood in the following discourse. I do not define time, space, place and
motion, as being well known to all. Only I must observe, that the vulgar
conceive those quantities under no other notions but from the relation they
bear to sensible objects. And thence arise certain prejudices, for the re
moving of which, it will be convenient to distinguish them into absolute
and relative, true and apparent, mathematical and common.

I. Absolute, true, and mathematical time, of itself, and from its own na
ture flows equably without regard to anything external, and by another
name is called duration : relative, apparent, and common time, is some sen
sible and external (whether accurate or unequable) measure of duration by
the means of motion, which is commonly used instead of true time ; such
as an hour, a day, a month, a year.

II. Absolute space, in its own nature, without regard to anything exter
nal, remains always similar and immovable. Relative space is some mo
vable dimension or measure of the absolute spaces ; which our senses de
termine by its position to bodies ; and which is vulgarly taken for immo
vable space ; such is the dimension of a subterraneous, an aereal, or celestial
space, determined by its position in respect of the earth. Absolute and
relative space, are the same in figure and magnitude ; but they do not re
main always numerically the same. For if the earth, for instance, moves,
a space of our air, which relatively and in respect of the earth remains al
ways the same, will at one time be one part of the absolute space into which

TS THE MATHEMATICAL PRINCIPLES

the air passes ; at another time it will be another part of the same, and so.
absolutely understood, it will be perpetually mutable.

III. Place is a part of space which a body takes up, and is according to
the space, either absolute or relative. I say, a part of space ; not the situation,
nor the external surface of the body. For the places of equal solids are
always equal ; but their superfices, by reason of their dissimilar figures, are
often unequal. Positions properly have no quantity, nor are they so much
the places themselves, as the properties of places. The motion of the whole
is the same thing with the sum of the motions of the parts ; that is, the
translation of the whole, out of its place, is the same thing with the sum
of the translations of the parts out of their places ; and therefore the place
of the whole is the same thing with the sum of the places of the parts, and
for that reason, it is internal, and in the whole body.

IV. Absolute motion is the translation of a body from one absolute
place into another ; and relative motion, the translation from one relative
place into another. Thus in a ship under sail, the relative place of a body
is that part of the ship which the body possesses ; or that part of its cavity
which the body fills, and which therefore moves together with the ship :
and relative rest is the continuance of the body in the same part of the
ship, or of its cavity. But real, absolute rest, is the continuance of the
body in the same part of that immovable space, in which the ship itself,
its cavity, and all that it contains, is moved. Wherefore, if the earth is
really at rest, the body, which relatively rests in the ship, will really and
absolutely move with the same velocity which the ship has on the earth.
But if the earth also moves, the true and absolute motion of the body will
arise, partly from the true motion of the earth, in immovable space ; partly
from the relative motion of the ship on the earth ; and if the body moves
also relatively in the ship ; its true motion will arise, partly from the true
motion of the earth, in immovable space, and partly from the relative mo
tions as well of the ship on the earth, as of the body in the ship ; and from
these relative motions will arise the relative motion of the body on the
earth. As if that part of the earth, where the ship is, was truly moved
toward the east, with a velocity of 10010 parts; while the ship itself, with
a fresh gale, and full sails, is carried towards the west, with a velocity ex
pressed by 10 of those parts ; but a sailor walks in the ship towards the
east, with 1 part of the said velocity ; then the sailor will be moved truly
in immovable space towards the east, with a velocity of 10001 parts, and
relatively on the earth towards the west, with a velocity of 9 of those parts.

Absolute time, in astronomy, is distinguished from relative, by the equa
tion or correction of the vulgar time. For the natural days are tr^y un
equal, though they are commonly considered as equal, and used for a meas
ure of time ; astronomers correct this inequality for their more accurate
deducing of the celestial motions. It may be, that there is no such thing
as an equable motion, whereby time may H accurately measured. All mo

OF NATURAL PHILOSOPHY. 79

tions may be accelerated and retarded; but the true, or equable, progress of
absolute time is liable to no change. The duration or perseverance of the
existence of things remains the same, whether the motions are swift or slow,
or none at all : and therefore it ought to be distinguished from what are
only sensible measures thereof ; and out of which we collect it, by means
of the astronomical equation. The necessity of which equation, for deter
mining the times of a phamomenon, is evinced as well from the experiments
of the pendulum clock, as by eclipses of the satellites of Jupiter.

As the order of the parts of time is immutable, so also is the order of
the parts of space. Suppose those parts to be moved out of their places, and
they will be moved (if the expression may be allowed) out of themselves.
For times and spaces are, as it were, the places as well of themselves as of
all other things. All things are placed in time as to order of succession ;
and in space as _to order of situation. It is from their essence or nature
that they are places ; and that the primary places of things should be
moveable, is absurd. These are therefore the absolute places ; and trans
lations out of those places, are the only absolute motions.

But because the parts of space cannot be seen, or distinguished from one
another by our senses, therefore in their stead we use sensible measures of
them. For from the positions and distances of things from any body con
sidered as immovable, we define all places ; and then with respect to such
places, we estimate all motions, considering bodies as transferred from some
of those places into others. And'so, instead of absolute places and motions,
we use relative ones; and that without any inconvenience in common af
fairs ; but in philosophical disquisitions, we ought to abstract from our
senses, and consider things themselves, distinct from what are only sensible
measures of them. For it may be that there is no body really at rest, to
which the places and motions of others may be referred.

But we may distinguish rest and motion, absolute and relative, one from
the other by their properties, causes and effects. It is a property of rest,
that bodies really at rest do rest in respect to one another. And therefore
as it is possible, that in the remote regions of the fixed stars, or perhaps
far beyond them, there may be some body absolutely at rest ; but impossi
ble to know, from the position of bodies to one another in our regions
whether any of these do keep the same position to that remote body; it
follows that absolute rest cannot be determined from the position of bodies
in our regions.

It is a property of motion, that the parts, which retain given positions
to their wholes, do partake of the motions of those wholes. For all the
parts of revolving bodies endeavour to recede from the axis of motion ;
and the impetus of bodies moving forward, arises from the joint impetus
of all the parts. Therefore, if surrounding bodies are moved, those that
are relatively at rest within them, will partake of their motion. Upon
which account, the true and absolute motion of a body cannot be Jeter-

8C THE MATHEMATICAL PRINCIPLES

mined by the translation of it from those which only seem to rest ; for the
external bodies ought not only to appear at rest, but to be really at rest.
For otherwise, all included bodies, beside their translation from near the
surrounding ones, partake likewise of their true motions ; and though that
translation were not made they would not be really at rest, but only seem
to be so. For the surrounding bodies stand in the like relation to the
surrounded as the exterior part of a whole does to the interior, or as the
shell does to the kernel ; but, if the shell moves, the kernel will also
move, as being part of the whole, without any removal from near the shell.

A property, near akin to the preceding, is this, that if a place is moved,
whatever is placed therein moves along with it ; and therefore a body,
which is moved from a place in motion, partakes also of the motion of its
place. Upon which account, all motions, from places in motion, are no
other than parts of entire and absolute motions ; and every entire motion
is composed of the motion of the body out of its first place, and the
motion of this place out of its place ; and so on, until we come to some
immovable place, as in the before-mentioned example of the sailor. Where
fore, entire and absolute motions can be no otherwise determined than by
immovable places : and for that reason I did before refer those absolute
motions to immovable places, but relative ones to movable places. Now
no other places are immovable but those that, from infinity to infinity, do
all retain the same given position one to another ; and upon this account
must ever remain unmoved ; and do thereby constitute immovable space.

The causes by which true and relative motions are distinguished, one
from the other, are the forces impressed upon bodies to generate motion.
True motion is neither generated nor altered, but by some force impressed
upon the body moved : but relative motion may be generated or altered
without any force impressed upon the body. For it is sufficient only to
impress some force on other bodies with which the former is compared,
that by their giving way, that relation may be changed, in which the re
lative rest or motion of this other body did consist. Again, true motion
suffers always some change from any force impressed upon the moving
body ; but relative motion docs not necessarily undergo any change by such
forces. For if the same forces are likewise impressed on those other bodies,
with which the comparison is made, that the relative position may be pre
served, then that condition will be preserved in which the relative motion
consists. And therefore any relative motion may be changed when the
true motion remains unaltered, and the relative may be preserved when the
true suffers some change. Upon which accounts; true motion does by no
means consist in such relations.

The effects whicli distinguish absolute from relative motion arc, the
forces of receding from the axis of circular motion. For there are no such
forces in a circular motion purely relative, but in a true and absolute cir
cular motion., they are greater or less, according t » the quantity of the

OF NATURAL PHILOSOPHY. «1

motion. If a vessel, hung: by & }ong cord, is so often turned ubout that the
cord is strongly twisted, then filled with water, and held at rest together
with the water ; after, by the sudden action of another force, it is whirled
about the contrary way, and while the cord is untwisting itself, the vessel
continues for some time in this motion ; the surface of the water will at
first be plain, as before the vessel began to move : but the vessel; by grad
ually communicating its motion to the water, will make it begin sensibly
^to revolve, and recede by little and little from the middle, and ascend to the
sides of the vessel, forming itself into a concave figure (as I have experi
enced), and the swifter the motion becomes, the higher will the water rise,
till at last, performing its revolutions in the same times with the vessel,
it becomes relatively at rest in it. This ascent of the water shows its en
deavour to recede from the axis of its motion ; and the true and absolute
circular motion of the water, which is here directly contrary to the rela-
tivej discovers itself, and may be measured by this endeavour. At first,
when the relative motion of the water in the vessel was greatest, it pro
duced no endeavour to recede from the axis ; the water showed no tendency
to the circumference, nor any ascent towards the sides of the vessel, but
remained of a plain surface, and therefore its true circular motion had not
yet begun. But afterwards, when the relative motion of the water had
decreased, the ascent thereof towards the sides of the vessel proved its en
deavour to recede from the axis ; and this endeavour showed the real cir
cular motion of the water perpetually increasing, till it had acquired its
greatest quantity, wh en the water rested relatively in the vessel. And
therefore this endeavour does not depend upon any translation of the water
in respect of the ambient bodies, nor can true circular motion be defined
by such translation. There is only one real circular motion of any one
revolving body, corresponding to only one power of endeavouring to recede
from its axis of motion, as its proper and adequate effect ; but relative
motions, in one and the same body, are innumerable, according to the various
relations it bears to external bodies, and like other relations, arc altogether
destitute of any real effect, any otherwise than they may perhaps par
take of that one only true motion. And therefore in their system who
suppose that our heavens, revolving below the sphere of the fixed stars,
carry the planets along with them ; the several parts of those heavens, and
the planets, which are indeed relatively at rest in their heavens, do yet
really move. For they change their position one to another (which never
happens to bodies truly at rest), and being carried together with their
heavens, partake of their motions, and as parts of revolving wholes,
endeavour to recede from the axis of their motions.

Wherefore relative quantities are not the quantities themselves, whose
names they bear, but those sensible measures of them (either accurate cr
inaccurate), which arc commonly used instead of the measured quantities
themselves. And if the meaning of words is to he determined bv their

82 THE MATHEMATICAL PRINCIPLES

use, then by the names time, space, place and motion, their measures arv
properly to be understood ; and the expression will be unusual, and purely
mathematical, if the measured quantities themselves are meant. Upon
which account, they do strain the sacred writings, who there interpret
those words for the measured quantities. Nor do those less defile the
purity of mathematical and philosophical truths, who confound real quan
tities themselves with their relations and vulgar measures.

It is indeed a matter of great difficulty to discover, and effectually to
distinguish, the true motions of particular bodies from the apparent ; be
cause the parts of that immovable space, in which those motions are per
formed, do by no means come under the observation of our senses. Yet
the thing is not altogether desperate : for we have some arguments to
guide us, partly from the apparent motions, which are the differences of
the true motions ; partly from the forces, which are the causes and effects
of the true motions. For instance, if tAvo globes, kept at a given distance
one from the other by means of a cord that connects them, were revolved
about their common centre of gravity, we might, from the tension of the
cord, discover the endeavour of the globes to recede from the axis of their
motion, and from thence we might compute the quantity of their circular
motions. And then if any equal forces should be impressed at once on the
alternate faces of the globes to augment or diminish their circular motions,
from the increase or decr» ase of the tensicn of 1 le cord, we might infer
the increment or decrement of their motions : and thence would be found
on what faces those forces ought to be impressed, that the motions of the
globes might be most augmented ; that is, we might discover their hinder-
most faces, or those which, in the circular motion, do follow. But the
faces which follow being known, and consequently the opposite ones that
precede, we should likewise know the determination of their motions. And
thus we might find both the quantity and the determination of this circu
lar motion, even in an immense vacuum, where there was nothing external
or sensible with which the globes could be compared. But now, if in that
space some remote bodies were placed that kept always a given position
one to another, as the fixed stars do in our regions, we could not indeed
determine from the relative translation of the globes among those bodies,
whether the motion did belong to the globes or to the bodies. But if we
observed the cord, and found that its tension was that very tension which
the motions of the globes required, we might conclude the motion to be in
the globes, and the bodies to be at rest ; and then, lastly, from the trans
lation of the globes among the bodies, we should find the determination oi
their motions. But how we are to collect the true motions from their
causes, effects, and apparent differences ; and, vice versa, how from the mo
tions, either true or apparent, we may come to the knowledge of theii
causes and effects, shall be explained more at large in the following tra<;t
For to this end it was that I composed it.

OF NATURAL PHILOSOPHY.

AXIOMS, OR LAWS OF MOTION.

LAW I.

Hvery body perseveres in its state of rest, or of uniform motion in a ri^ht
line, unless it is compelled to change that state by forces impressed
thereon.

PROJECTILES persevere in their motions, so far as they are not retarded
by the resistance of the air, or impelled downwards by the force of gravity
A top, whose parts by their cohesion are perpetually drawn aside from
rectilinear motions, does not cease its rotation, otherwise than as it is re
tarded by the air. The greater bodies of the planets and comets, meeting
with less resistance in more free spaces, preserve then jDotions both pro
gressive and circular for a much longer time.

LAW II.

The alteration of motion is ever proportional to the motive force imp reus
ed ; and is made in the direction of the right line in. which that force
is impressed.

If any force generates a motion, a double force will generate double the
motion, a triple force triple the motion, whether that force be impressed
altogether and at once, or gradually and successively. And this motion
(being always directed the same way with the generating force), if the body
moved before, is added to or subducted from the former motion, according
as they directly conspire with or are directly contrary to each other ; or
obliquely joined, when they are oblique, so as to produce a new motion
compounded from the determination of both.

LAW III.

To every action there is always opposed an equal reaction : or the mu
tual actions of two bodies upon each other are always equal, and di
rected to contrary parts.

Whatever draws or presses another is as much drawn or pressed by that
other. If you press a stone with your finger, the finger is also pressed by
the stone. If a horse draws a stone tied to a rope, the horse (if I may so
say) will be equally drawn back towards the stone: for the distended rope,
by the same endeavour to relax or unbend itself, will draw the horse as
much towards the stone, as it does the stone towards the horse, and will
obstruct the progress of the one as much as it advances that of the other.

84 THE MATHEMATICAL PRINCIPLES

If a body impinge upon another, and by its force change the motion of (It*
other, that body also (because of the equality of the mutual pressure) will
undergo an equal change, in its own motion, towards the contrary part.
The changes made by these actions are equal, not in the velocities but in
the motions of bodies ; that is to say, if the bodies are not hindered by any
other impediments. For, because the motions are equally changed, the
changes of the velocities made towards contrary parts are reciprocally pro
portional to the bodies. This law takes place also in attractions, as will
be proved in the next scholium.

COROLLARY I.

A body by two forces conjoined will describe the diagonal of a parallelo
gram, in the same time that it wovld describe the sides, by those forces
apart.

If a body in a given time, by the force M impressed
apart in the place A, should with an uniform motion /
be carried from A to B ; and by the force N impressed
apart in the same place, should be carried from A to c ~\)
C ; complete the parallelogram ABCD, and, by both forces acting together,
it will in the same time be carried in the diagonal from A to D. For
since the force N acts in the direction of the line AC, parallel to BD, this
force (by the second law) will not at all alter the velocity generated by the
other force M, by which the body is carried towards the line BD. The
body therefore will arrive at the line BD in the same time, whether the
rorce N be impressed or not ; and therefore at the end of that time it will
he found somewhere in the line BD. By the same argument, at the end
of the same time it AY ill be found somewhere in the line CD. Therefore it
will be found in the point D, where both lines meet. But it will move in
;i right line from A to D, by Law I.

COROLLARY II.

And hence is explained the composition of any one direct force AD, out
of any two oblique forces AC and CD ; and, on the contrary, the re
solution of any one direct force AD into two oblique forces AC and
CD : which composition and resolution are abundantly confirmed from,
mechanics.

As if the unequal radii OM and ON drawn from the centre O of any
wheel, should sustain the weights A and P by the cords MA and NP ; and
the forces of those weights to move the wheel were required. Through the
rentre O draw the right line KOL, meeting the cords perpendicularly in
A and L; and from the centre O, with OL the greater of the distances

OF NATURAL PHILOSOPHY.

OK arid OL, describe a circle, meeting the cord
MA in D : and drawing OD, make AC paral- "^
lei and DC perpendicular thereto. Now, it
being indifferent whether the points K, L, D, of
the cords be lixed to the plane of the wheel or
not, the weights will have the same effect
whether they are suspended from the points K
and L, or from D and L. Let the whole force
of the weight A be represented by the line AD,
and let it be resolved into the forces AC and
CD ; of which the force AC, drawing the radius
OD directly from the centre, will have no effect to move the wheel : but
the other force DC, drawing the radius DO perpendicularly, will have the
same effect as if it drew perpendicularly the radius OL equal to OD ; that
is, it w ill have the same effect as the weight P, if that weight is to the
weight A as the force DC is to the force DA ; that is (because of the sim
ilar triangles ADC, DOK), as OK to OD or OL. Therefore the weights A
and P, which are reciprocally as the radii OK and OL that lie in the same
right line, will be equipollent, and so remain in equilibrio ; which is the well
known property of the balance, the lever, and the wheel. If either weight is
greater than in this ratio, its force to move the wheel will be so much greater.

If the weight p, equal to the weight P, is partly suspended by the
cord NJO, partly sustained by the oblique plane pG ; draw p}i, NH, the
former perpendicular to the horizon, the latter to the plane pG ; and if
the force of the weight p tending downwards is represented by the line
/?H, it may be resolved into the forces joN, HN. If there was any plane
/?Q, perpendicular to the cord y?N, cutting the other plane pG in a line
parallel to the horizon, and the weight p was supported only by those
planes pQ, pG, it would press those planes perpendicularly with the forces
pN, HN; to wit, the plane joQ, with the force joN, and the plane pG with
the force HN. And therefore if the plane pQ was taken away, so thnt
the weight might stretch the cord, because the cord, now sustaining the
weight, supplies the place of the plane that was removed, it will be strained
by the same force joN which pressed upon the plane before. Therefore,
the tension of this oblique cord joN will be to that of the other perpendic
ular cord PN as jt?N to joH. And therefore if the weight p is to the
weight A in a ratio compounded of the reciprocal ratio of the least distances
of the cords PN, AM, from the centre of the wheel, and of the direct ratio of
pH tojoN, the weights will have the same effect towards moving the wheel,
and will therefore sustain each other : as any one may find by experiment.

But the weight p pressing upon those two oblique planes, may be con
sidered as a wedge between the two internal surfaces of a body split by it;
and hence tlif ft IV.P* of th^ v, ^dge and the mallet may be determined; foi

8G THE MATHEMATICAL PRINCIPLES

because the force with which the weight p presses the plane pQi is to the
force with which the same, whether by its own gravity, or by the blow of
a mallet, is impelled in the direction of the line joH towards both the
planes, as joN to pH ; and to the force with which it presses the other
plane pG, as joN to NH. And thus the force of the screw may be deduced
from a like resolution of forces ; it being no other than a wedge impelled
with the force of a lever. Therefore the use of this Corollary spreads far
and wide, and by that diffusive extent the truth thereof is farther con
firmed. For on what has been said depends the whole doctrine of mechan
ics variously demonstrated by different authors. For from hence are easily
deduced the forces of machines, which are compounded of wheels, pullics,
levers, cords, and weights, ascending directly or obliquely, and other mechan
ical powers ; as also the force of the tendons to move the bones of animals.

COROLLARY III.

The (/uaittity of motion, which is collected by taking the sum of the mo
tions directed towards the same parts, and the difference of those that
are directed to contrary parts, suffers no change from the action oj
bodies among themselves.

For action and its opposite re-action are equal, by Law III, and there
fore, by Law II, they produce in the motions equal changes towards oppo
site parts. Therefore if the motions are directed towards the same parts.
whatever is added to the motion of the preceding body will be subducted
from the motion of that which follows ; so that the sum will be the same
as before. If the bodies meet, with contrary motions, there will be an
equal deduction from the motions of both ; and therefore the difference of
the motions directed towards opposite parts will remain the same.

Thus if a spherical body A with two parts of velocity is triple of a
spherical body B which follows in the same right line with ten parts of
velocity, the motion of A will be to that of B as 6 to 10. Suppose,
then, their motions to be of 6 parts and of 10 parts, and the sum will be
16 parts. Therefore, upon the meeting of the bodies, if A acquire 3, 4,
or 5 parts of motion, B will lose as many ; and therefore after reflexion
A will proceed With 9, 10, or 11 parts, and B with 7, 6, or 5 parts; the
sum remaining always of 16 parts as before. If the body A acquire 9,
10, 11, or 12 parts of motion, and therefore after meeting proceed with
15, 16, 17, or 18 parts, the body B, losing so many parts as A has got,
will either proceed with 1 part, having lost 9, or stop and remain at rest,
as having lost its whole progressive motion of 10 parts ; or it will go back
with 1 part, having not only lost its whole motion, but (if 1 may so say)
one part more; or it will go back with 2 parts, because a progressive mo
tion of 12 parts is taken off. And so the sums of the Conspiring motions
15 ,1, or 16-1-0, and the differences of the contrary i otions 17 — 1 and

OF NATURAL PHILOSOPHY.

[S — 2, will always be equal to 16 parts, as they were before tie meeting
and reflexion of the bodies. But, the motions being known with whicli
the bodies proceed after reflexion, the velocity of either will be also known,
by taking the velocity after to the velocity before reflexion, as the motion
after is to the motion before. As in the last case, where the motion of tho
body A was of 0 parts before reflexion and of IS parts after, and the
velocity was of 2 parts before reflexion, the velocity thereof after reflexion
will be found to be of 6 parts ; by saying, as the 0 parts of motion before
to 18 parts after, so are 2 parts of velocity before reflexion to (5 parts after.
But if the bodies are cither not spherical, or, moving in different right
lines, impinge obliquely one upon the other, and their mot'ons after re
flexion are required, in those cases we are first to determine the position
of the plane that touches the concurring bodies in the point of concourse ,
then the motion of each body (by Corol. II) is to be resolved into two, one
perpendicular to that plane, and the other parallel to it. This done, be
cause the bodies act upon each other in the direction of a line perpendicu
lar to this plane, the parallel motions are to be retained the same after
reflexion as before ; and to the perpendicular motions we are to assign
equal changes towards the contrary parts ; in such manner that the sum
of the conspiring and the difference of the contrary motions may remain
the same as before. From such kind of reflexions also sometimes arise
the circular motions of bodies about their own centres. But these are
cases which I do not consider in what follows ; and it would be too tedious
to demonstrate every particular that relates to this subject.

COROLLARY IV.

The common centre of gravity of two or more bodies does not alter its
state of 'motion or rest by the actions of the bodies among themselves ;
and therefore the common centre of gravity of all bodies acting upon
each other (excluding outward actions and impediments) is either at
rest, or moves uniformly in a right line.

For if two points proceed with an uniform motion in right lines, and
their distance be divided in a given ratio, the dividing point will be either
at rest, or proceed uniformly in a right line. This is demonstrated here
after in Lem. XXIII and its Corol., when the points are moved in the same
plane ; and by a like way of arguing, it may be demonstrated when the
points are not moved in the same plane. Therefore if any number of
Kdies move uniformly in right lines, the common centre of gravity of any
two of them is either at rest, or proceeds uniformly in a right line ; because
the line which connects the centres of those two bodies so moving is divided at
that common centre in a given ratio. In like manner the common centre
of those two and that of a third body will be either at rest or moving uni
formly in aright line because at that centre the distance 1 etween th«?

88 THE MATHEMATICAL PRINCIPLES

common centre of the two bodies, and the centre of this last, is divided in
a given ratio. In like manner the common centre of these three, and of a
fourth body, is either at rest, or moves uniformly in a right line ; because
the distance between the common centre of the three bodies, and the centre
of the fourth is there also divided in a given ratio, and so on m itifinitum.
Therefore, in a system of bodies where there is neither any mutual action
among themselves, nor any foreign force impressed upon them from without,
and which consequently move uniformly in right lines, the common centre of
gravity of them all is either at rest or moves uniformly forward in a right line.
Moreover, in a system of two bodies mutually acting upon each other,
since the distances between their centres and the common centre of gravity
of both are reciprocally as the bodies, the relative motions of those bodies,
whether of approaching to or of receding from that centre, will be equal
among themselves. Therefore since the changes which happen to motions
are equal and directed to contrary parts, the common centre of those bodies,
by their mutual action between themselves, is neither promoted nor re
tarded, nor suffers any change as to its state of motion or rest. But in a
system of several bodies, because the common centre of gravity of any two
acting mutually upon each other suffers no change in its state by that ac
tion : and much less the common centre of gravity of the others with which
that action does not intervene ; but the distance between those two centres
is divided by the common centre of gravity of all the bodies into parts re
ciprocally proportional to the total sums of those bodies whose centres they
are : and therefore while those two centres retain their state of motion or
rest, xhe common centre of all does also retain its state : it is manifest that
the common centre of all never suffers any change in the state of its mo
tion or rest from the actions of any two bodies between themselves. But
in such & system all the actions of the bodies among themselves either hap
pen between two bodies, or are composed of actions interchanged between
some two bodies ; and therefore they do never produce any alteration in
the comrrv n centre of alias to its state of motion or rest. Wherefore
tiince that centre, when the bodies do not act mutually one upon another,
Oilier is nt rest or moves uniformly forward in some right line, it will,
:v\>U7ithst?nding the mutual actions of the bodies among themselves, always
jAY-jevere in its state, either of rest, or of proceeding uniformly in a right
liiv,, unless it is forced out of this state by the action of some power im-
prev^-d from without upon the whole system. And therefore the same law
take*1 place in a system consisting of many bodies as in one single body,
with wsgard to their persevering in their state of motion or of rest. For
the pi \\jressive motion, whether of one single body, or of a whole system of
bodies us always to be estimated from the motion of the centre of gravity.

COROLLARY V.

The motions cf bcdies included in a given space a ~e Ike same among

OF NATURAL PHILOSOPHY. 89

themselves, whether that space is at rest, or moves uniformly forwards

in a right line without any circular motion.

For the differences of the motions tending towards the same parts, and
the sums of those that tend towards contrary parts, are, at first (by sup
position), in both cases the same ; and it is from those sums and differences
that the collisions and impulses do arise with which the bodies mutually
impinge one upon another. Wherefore (by Law II), the effects of those
collisions will be equal in both cases ; and therefore the mutual motions
of the bodies among themselves in the one case will remain equal to the
mutual motions of the bodies among themselves in the other. A clear
proof of which we have from the experiment of a ship ; where all motions
happen after the same manner, whether the ship is at rest, or is carried
uniformly forwards in a right line.

COROLLARY VI.

If bodies, any how moved among themselves, are urged in the direct-ton
of parallel lines by equal accelerative forces, they will all continue to
move among themselves, after the same manner as if they had been
urged by no such forces.

For these forces acting equally (with respect to the quantities of the
DO dies to be moved), and in the direction of parallel lines, will (by Law II)
move all the bodies equally (as to velocity), and therefore will never pro
duce any change in the positions or motions of the bodies among themselves.

SCHOLIUM.

Hitherto I have laid down such principles as have been received by math
ematicians, and are confirmed by abundance of experiments. By the first
two Laws and the first two Corollaries, Galileo discovered that the de
scent of bodies observed the duplicate ratio of the time, and that the mo
tion of projectiles was in the curve of a parabola; experience agreeing
with both, unless so far as these motions are a little retarded by the re
sistance of the air. When a body is falling, the uniform force of its
gravity acting equally, impresses, in equal particles of time, equal forces
upon that body, and therefore generates equal velocities; and in the whole
time impresses a whole force, and generates a whole velocity proportional
to the time. And the spaces described in proportional times are as the
velocities and the times conjunctly ; that is, in a duplicate ratio of the
times. And when a body is thrown upwards, its uniform gravity im
presses forces and takes off velocities proportional to the times ; and the
times of ascending to the greatest heights are as the velocities to be taken
off, and those heights are as the velocities and the times conjunetly, or ir,
the duplicate ratio of the velocities. And if a body be projected in any
direction, the motion arising from its projection jS compounded with the

90

THE MATHEMATICAL PRINCIPLES

motion arising from its gravity. As if the body A by its motion of pio-
jection alone could describe in a given time the right line
AB, and with its motion of falling alone could describe in
the same time the altitude AC ; complete the paralello-
gram ABDC, and the body by that compounded motion
will at the end of the time be found in the place D ; and
the curve line AED, which that body describes, will be a
parabola, to which the right line AB will be a tangent in
A ; and whose ordinate BD will be as the square of the line AB. On the
same Laws and Corollaries depend those things which have been demon
strated concerning the times of the vibration of pendulums, and are con
firmed by the daily experiments of pendulum clocks. By the same, to
gether with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huv-
gens, the greatest geometers of our times, did severally determine the rules
of the congress and reflexion of hard bodies, and much about the same
time communicated their discoveries to the Royal Society, exactly agreeing
among themselves as to those rules. Dr. Wallis, indeed, was something
more early in the publication ; then followed Sir Christopher Wren, and,
lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of
the thing before the Royal Society by the experiment of pendulums, which
Mr. Mariottc soon after thought fit to explain in a treatise entirely upon
that subject. But to bring this experiment to an accurate agreement with
the theory, we are to have a due regard as well to the resistance of the air

bodies.

Let the spherical bodies
CD F II

as to the clastic force of the concurrin
A, B be suspended by the parallel and
equal strings AC, Bl), from the centres
C, D. About these centres, with those
intervals, describe the semicircles EAF,
GBH, bisected by the radii CA, DB.
Bring the body A to any point R of the
arc EAF, and (withdrawing the body
B) let it go from thence, and after one oscillation suppose it to return to
the point V : then RV will be the retardation arising from the resistance
of the air. Of this RV let ST be a fourth part, situated in the middle.
to wit, so as RS and TV may be equal, and RS may be to ST as 3 to 2
then will ST represent very nearly the retardation during the descent
from S to A. Restore the body B to its place: and, supjx sing the body
A to be let fall from the point S, the velocity thereof in the place of re
flexion A, without sensible error, will be the same as if it had descended
m vacit.o from the point T. Upon which account this velocity may be
represented by the chord of the arc TA. For it is a proposition well
known to geometers, that the velocity of a pendulous body in the loAvest
point is as the chord of the arc which it has described in its descent. Aftci

OF NATUltAL PHILOSOPHY. 9 I

reflexion, suppose the body A comes to the place s, and the body B to the
place k. Withdraw the body B, and find the place v, from which if the
body A, being let go, should after one oscillation return to the place r, st
may be a fourth part of rv. so placed in the middle thereof as to leave is
equal to tv, and let the chord of the arc tA represent the velocity which
the body A had in the place A immediately after reflexion. For t will be
the true and correct place to which the body A should have ascended, if
the resistance of the air had been taken off. In the s.ime way we are to
correct the place k to which the body B ascends, by finding the place I to
which it should have ascended in vacuo. And thus everything may be
subjected to experiment, in the same manner as if we were really placed
in vacuo. These things being done, we are to take the product (if I may
so say) of the body A, by the chord of the arc TA (which represents its
velocity), that we may have its motion in the place A immediately before
reflexion ; and then by the chord of the arc /A, that we may have its mo
tion in the place A immediately after reflexion. And so we are to take
the product of the body B by the chord of the arc B/, that we may have
the motion of the same immediately after reflexion. And in like manner,
when two bodies are let go together from different places, we are to find
the motion of each, as well before as after reflexion; and then we may
compare the motions between themselves, and collect the effects of the re
flexion. Thus trying the thing with pendulums of ten feet, in unequal
as well as equal bodies, and making the bodies to concur after a descent
through large spaces, as of 8, 12, or 16 feet, I found always, without an
error of 3 inches, that when the bodies concurred together directly, equal
changes towards the contrary parts were produced in their motions, and,
of consequence, that the action and reaction were always equal. As if the
body A impinged upon the body B at rest with 9 parts of motion, and
losing 7, proceeded after reflexion with 2, the body B was carried back
wards with those 7 parts. If the bodies concurred with contrary motions,
A with twelve parts of motion, and B with six, then if A receded with J4,
B receded with 8 ; to wit, with a deduction of 14 parts of motion on
each side. For from the motion of A subducting twelve parts, nothing
will remain ; but subducting 2 parts more, a motion will be generated of
2 parts towards the contrary way ; and so, from the motion of the body
B of 6 parts, subducting 14 parts, a motion is generated of 8 parts towards
the contrary way. But if the bodies were made both to move towards the
same way, A, the swifter, with 14 parts of motion, B, the slower, with 5,
and after reflexion A went on with 5, B likewise went on with 14 parts ;
9 parts being transferred from A to B. And so in other cases. By the
congress and collision of bodies, the quantity of motion, collected from the
sum of the motions directed towards the same way, or from the difference,
of those that were directed towards contrary ways, was never changed.
For the error of an inch or two in measures may be easily ascribed to tht

92 THE MATHEMATICAL PRINCIPLES

difficulty of executing everything with accuracy. It was not easy to let
go the two pendulums so exactly together that the bodies should impinge
one upon the other in the lowermost place AB ; nor to mark the places s,
and ky to which the bodies ascended after congress. Nay, and some errors,
too, might have happened from the unequal density of the parts of the pen
dulous bodies themselves, and from the irregularity of the texture pro
ceeding from other causes.

But to prevent an objection that may perhaps be alledged against the
rule, for the proof of which this experiment was made, as if this rule did
suppose that the bodies were either absolutely hard, or at least perfectly
elastic (whereas no such bodies are to be found in nature), 1 must add. that
the experiments we have been describing, by no means depending upon
that quality of hardness, do succeed as well in soft as in hard bodies. For
if the rule is to be tried in bodies not perfectly hard, we are only to di
minish the reflexion in such a certain proportion as the quantity of the
elastic force requires. By the theory of Wren and Huygens, bodies abso
lutely hard return one from another with the same velocity with which
they meet. But this may be affirmed with more certainty of bodies per
fectly elastic. In bodies imperfectly elastic the velocity of the return is to
be diminished together with the elastic force ; because that force (except
when the parts of bodies are bruised by their congress, or suffer some such
extension as happens under the strokes of a hammer) is (as far as I can per
ceive) certain and determined, and makes the bodies to return one from
the other with a relative velocity, which is in a given ratio to that relative
velocity with which they met. This I tried in balls of wool, made up
tightly, and strongly compressed. For, first, by letting go the pendulous
bodies, and measuring their reflexion, I determined the quantity of their
elastic force ; and then, according to this force, estimated the reflexions
that ought to happen in other cases of congress. And with this computa
tion other experiments made afterwards did accordingly agree ; the balls
always receding one from the other with a relative velocity, which was to
the relative velocity with which they met as about 5 to 9. Balls of steel
returned with almost the same velocity : those of cork with a velocity some-^
thing less ; but in balls of glass the proportion was as about 15 to 16.
And thus the third Law, so far as it regards percussions and reflexions, is
proved by a theory exactly agreeing with experience.

In attractions, I briefly demonstrate the thing after this manner. Sup
pose an obstacle is interposed to hinder the congress of any two bodies A.
B, mutually attracting one the other : then if either body, as A, is more
attracted towards the other body B, than that other body B is towards the
first body A, the obstacle will be more strongly urged by the pressure of
the body A than by the pressure of the body B, and therefore will not
remain in equilibrio : but the stronger pressure will prevail, and will make
the system of the two bodies, together with the obstacle, to move directly

OF NATURAL PHILOSOPHY.

93

towards the parts on which B lies ; arid in free spaces, to go forward in
infmitiim with a motion perpetually accelerated ; which is absurd and
contrary to the first Law. For, by the first Law, the system ought to per
severe in its state of rest, or of moving uniformly forward in a right line :
and therefore the bodies must equally press the obstacle, and be equally
attracted one by the other. I made the experiment on the loadstone and
iron. If these, placed apart in proper vessels, are made to float by one
another in standing water, neither of them will propel the other ; but,
by being equally attracted, they will sustain each other's pressure, and rest
at last in an equilibrium.

So the gravitation betwixt the earth and its parts is mutual. Let the
earth FI be cut by any plane EG into two parts EGF
and EGI, and their weights one towards the other
will be mutually equal. For if by another plane
HK, parallel to the former EG, the greater partFJ
EGI is cut into two parts EGKH and HKI.
whereof HKI is equal to the part EFG, first cut
oft', it is evident that the middle part EGKH, will
have no propension by its proper weight towards either side, but will hang
as it were, and rest in an equilibrium betwixt both. But the one extreme
part HKI will with its whole weight bear upon and press the middle part
towards the other extreme part EGF : and therefore the force with which
EGI, the sum of the parts HKI and EGKH, tends towards the third part
EGF, is equal to the weight of the part HKI, that is, to the weight of
the third part EGF. And therefore the weights of the two parts EGI
and EGF, one towards the other, are equal, as I was to prove. And in
deed if those weights were not equal, the whole earth floating in the non-
resisting aether would give way to the greater weight, and, retiring from
it, would be carried off in infinitum.

And as those bodies are equipollent in the congress and reflexion, whose
velocities are reciprocally as their innate forces, so in the use of mechanic
instruments those agents are equipollent, and mutually sustain each the
contrary pressure of the other, whose velocities, estimated according to the
determination of the forces, are reciprocally as the forces.

So those weights are of equal force to move the arms of a balance;
which during the play of the balance are reciprocally as their velocities
upw ards and downwards ; that is, if the ascent or descent is direct, those
weights are of equal force, which are reciprocally as the distances of the
points at which they are suspended from the axis oi the balance : but if
they are turned aside by the interposition of oblique planes, or other ob
stacles, and made to ascend or descend obliquely, those bodies will be
equipollent, wThich are reciprocally as the heights of their ascent and de
scent taken according to the perpendicular ; and that on account of the
determination of gravity downwards.

94 THE MATHEMATICAL PRINCIPLES

And in like manner in the pully, or in a combination of pullies, the
force of a hand drawing the rope directly, which is to the weight, whethel
ascending directly or obliquely, as the velocity of the perpendicular ascent
of the weight to the velocity of the hand that draws the rope, will sustain
the weight.

In clocks and such like instruments, made up from a combination of
wheels, the contrary forces that promote and impede the motion of the
wheels, if they are reciprocally as the velocities of the parts of the wheel
on which they are impressed, will mutually sustain the one the other.

The force of the screw to press a body is to the force of the hand that
turns the handles by which it is moved as the circular velocity of the
handle in that part where it is impelled by the hand is to the progressive
velocity of the screw towards the pressed body.

The forces by which the wedge presses or drives the two parts of the
wood it cleaves are to the force of the mallet upon the wedge as the pro-
press of the wedge in the direction of the force impressed upon it by the
mallet is to the velocity with which the parts of the wood yield to the
wedge, in the direction of lines perpendicular to the sides of the wedge.
And the like account is to be given of all machines.

The power and use of machines consist only in this, that by diminishing
the velocity we may augment the force, and the contrary : from whence
in all sorts of proper machines, we have the solution of this problem ; 7
move a given weight with a given power, or with a given force to over
come any other given resistance. For if machines are so contrived that the
velocities of the agent and resistant are reciprocally as their forces, the
agent will just sustain the resistant, but with a greater disparity of ve
locity will overcome it. So that if the disparity of velocities is so great
as to overcome all that resistance which commonly arises either from the
attrition of contiguous bodies as they slide by one another, or from the
cohesion of continuous bodies that are to be separated, or from the weights
of bodies to be raised, the excess of the force remaining, after all those re
sistances are overcome, will produce an acceleration of motion proportional
thereto, as well in the parts of {he machine as in the resisting body. But
to treat of mechanics is not my present business. I was only willing to
show by those examples the great extent and certainty of the third Law ot
motion. For if we estimate the action of the agent from its force and
velocity conjunctly, and likewise the reaction of the impediment conjuncth
from the velocities of its several parts, and from the forces of resistance
arising from the attrition, cohesion, weight, and acceleration of those parts,
the action and reaction 'YL the use of all sorts of machines will b" found
always equal to one another. And so far as the action is propagated by
the intervening instruments, and at last impressed upon tic resisting
body, the ultimate determination of the action will be always contrary to
the determination of the reaction.

OF NATURAL PHILOSOPHY

95

BOOK I.

OF THE MOTION OF BODIES.

SECTION I.

Of the method of first and last ratios of quantities, by the help wJicreoj
we demonstrate the propositions that follow.

LEMMA I.

Quantities, and the ratios of quantities, which in any finite time converge
continually to equality, and before the end of that time approach nearer
the one to the other than by any given difference, become ultimately
equal.

If you deny it, suppose them to be ultimately unequal, and let D be
their ultimate difference. Therefore they cannot approach nearer to
equality than by that given difference D ; which is against the supposition,

LEMMA II.

If in any figure AacE, terminated by the right (f
lines A a. AE, and the curve acE, there be in
scribed any number of parallelograms Ab, Be,
Cd, fyc., comprehended under equal bases AB,
BC, CD, ^«c., and the sides, Bb, Cc, Dd, ^c.,
parallel to one side Aa of the figure ; and the
parallelograms aKbl, bLcm, cMdn, §*c., are com
pleted. Then if the breadth of those parallelo- \
grams be supposed to be diminisJied, and their X BF C D |;
number to be augmented in infinitum : / say, that :he ultimate ratios
'which the inscribed fignre AKbLcMdD, the tin nmscribed figure
AalbmcndoE, and en rvilijiear figure AabcdE, will have to one another,
are ratios of equality.

For the difference of the inscribed and circumscribed figures is the sum
of the parallelograms K7, Lw, M//. Do. that is (from the equality of all
their bases), the rectangle under one of their bases K6 and the sum of their
altitudes Aa, that is, the rectangle ABla. But this rectangle, because

M

a

96

THE MATHEMATICAL PRINCIPLES

[BOOK 1

its breadth AB is supposed diminished • in infinitum, becomes less than
any given space. And therefore (by Lem. I) the figures inscribed and
circumscribed become ultimately equal one to the other; and much more
will the intermediate curvilinear figure be ultimately equal to either*
Q.E.D.

LEMMA III.

The same ultimate ratios are also ratios of equality, when the breadth^
AB, BC, DC, fyc., of the parallelograms are unequal, and are all di
minished in infinitum.

For suppose AF equal to the greatest breadth, and
complete the parallelogram FAaf. This parallelo
gram will be greater than the difference of the in
scribed and circumscribed figures ; but, because its
breadth AF is diminished in infinitum, it will be
come less than any given rectangle. Q.E.D.

COR. 1. Hence the ultimate sum of those evanes
cent parallelograms will in all parts coincide with

the curvilinear figure. A BF C D E

COR. 2. Much more will the rectilinear figure^comprehendcd under tne
chords of the evanescent arcs ab, be, cd, (fee., ultimately coincide with tl.c
curvilinear figure.

COR. 3. And also the circumscribed rectilinear figure comprehended
under the tangents of the same arcs.

COR. 4 And therefore these ultimate figures (as to their perimeters acE)
are not rectilinear, but curvilinear limi's of rectilinear figures.

LEMMA IV.

If in two figures AacE, PprT, you inscribe (as before)

two ranks of parallelograms, an equal number in

each rank, and, when their breadths are diminished

in infinitum. the ultimate ratios of the parallelograms

in one figure to those in the other, each to each respec
tively, are the same; I say, that those two figures

AacE, PprT, are to one another in that same ratio.

For as the parallelograms in the one are severally to p
the parallelograms in the other, so (by composition) is the <
sum of all in the one to the sum of all in the other : and
so is the one figure to the other; because (by Lem. Ill) the
former figure to the former sum, and the latter figure to the
latter sum, are both in the ratio of equality. Q.E.D.

COR. Hence if two quantities of any kind are any
how divided into an equal number of parts, and those A

£

SEC. I.] OF NATURAL PHILOSOPHY. 97

parts, when their number is augmented, and their magnitude diminished
in infinitum, have a given ratio one to the other, the first to the first, the
second to the second, and so on in order, the whole quantities will be one to
the other in that same given ratio. For if, in the figures of this Lemma,
the parallelograms are taken one to the other in the ratio of the parts, the
sum of the parts will always be as the sum of the parallelograms ; and
therefore supposing the number of the parallelograms and parts to be aug
mented, and their magnitudes diminished in infinitum, those sums will be
in the ultimate ratio of the parallelogram in the one figure to the corres
pondent parallelogram in the other ; that is (by the supposition), in the
ultimate ratio of any part of the one quantity to the correspondent part of
the other.

LEMMA V.

In similar figures, all sorts of homologous sides, whether curvilinear or
rectilinear, are proportional ; and the areas are in the duplicate ratio
of the homologous sides.

LEMMA VI.

If any arc ACB, given in position, is snb- _j
tended by its chord AB, and in any point
A, in the middle of the contiinied curva
ture, is touched by a right line AD, pro
duced both ways ; then if the points A R
and B approach one another and meet,
I say, the angle RAT), contained between,
the chord and the tangent, will be dimin- ?
ished in infinitum, a/id ultimately will vanish.

For if that angle does not vanish, the arc ACB will contain with the
tangent AD an angle equal to a rectilinear angle ; and therefore the cur
vature at the point A will not be continued, which is against the supposi
tion.

LEMMA VII.

The same things being supposed, I say that the ultimate ratio of the arc,

chord, and tangent, any one to any other, is the ratio of equality.

For while the point B approaches towards the point A, consider always

AB and AD as produced to the remote points b and d, and parallel to the

secant BD draw bd : and let the arc Acb be always similar to the arc

ACB. Then, supposing the points A and B to coincide, the angle dAb

will vanish, by the preceding Lemma; and therefore the right lines Ab,

Arf (which are always finite), and the intermediate arc Acb, will coincide,

and become equal among themselves. Wheref ,re, the right lines AB, AD,

98 THE MATHEMATICAL PRINCIPLES [SEC. I.

and the intermediate arc ACB (which are always proportional to the
former), will vanish, and ultimately acquire the ratio of equality. Q.E.D.

COR. 1. Whence if through B we draw A

BP parallel to the tangent, always cutting
any right line AF passing through A in F/— i-

P, this line BP will be ultimately in the

ratio of equality with the evanescent arc ACB ; because, completing the
parallelogram APBD, it is always in a ratio of equality with AD.

COR. 2. And if through B and A more right lines are drawn, as BE,
I5D, AF, AG, cutting the tangent AD and its parallel BP : the ultimate
ratio of all the abscissas AD, AE, BF, BG, and of the chord and arc AB,
any one to any other, will be the ratio of equality.

COR. 3. And therefore in all our reasoning about ultimate ratios, we
may freely use any one of those lines for any other.

LEMMA VIII.

If the right lines AR, BR, with the arc ACB, the chord AB, and the
tangent AD, constitute three triangles RAB. RACB, RAD, and the
points A and B approach and meet : I say, that the ultimate form oj
these evanescent triangles is that of similitude, and their ultimate
ratio that of equality.

For while the point B approaches towards A
the point A, consider always AB, AD, AR,
as produced to the remote points b, d, and r,
and rbd as drawn parallel to RD, and let
the arc Acb be always similar to the arc
ACB. Then supposing the points A and B
to coincide, the angle bAd will vanish ; and
therefore the three triangles rAb, rAcb,rAd
^which are always finite), will coincide, and on that account become both
similar and equal. And therefore the triangles RAB. RACB, RAD
which are always similar and proportional to these, will ultimately be
come both similar and equal among themselves. Q..E.D.

COR. And hence in all reasonings about ultimate ratios, we may indif
ferently use any one of those triangles for any other.

LEMMA IX.

If a ngnt line AE. and a curve tine ABC, both given by position, cut
each other in a given angle, A ; and to that right line, in another
given angle, BD, CE are ordinately applied, meeting the curve in B,
C : and the points B and C together approach towards and meet in
the point A : / say, that the areas of the triangles ABD, ACE, wilt
ultimately be one to the other in the duplicate ratio of the sides.

BOOK LI

OF NATURAL PHILOSOPHY.

For while the points B, C, approach
towards the point A, suppose always AD
to be produced to the remote points d and .
e, so as Ad, Ae may be proportional to
AD, AE ; and the ordinates db, ec, to be
drawn parallel to the ordinates DB and
EC, and meeting AB and AC produced D
in b and c. Let the curve A be be similar
to the curve A BC, and draw the right line
Ag- so as to touch both curves in A, and
cut the ordinates DB, EC, db ec, in F, G,
J] g. Then, supposing the length Ae to remain the same, let the points B
and C meet in the point A ; and the angle cAg vanishing, the curvilinear
areas AW, Ace will coincide with the rectilinear areas A/rf, Age ; and
therefore (by Lem. V) will be one to the other in the duplicate ratio of
the sides Ad, Ae. But the areas ABD, ACE are always proportional to
these areas ; and so the sides AD, AE are to these sides. And therefore
the areas ABD, ACE are ultimately one to the other in the duplicate ratio
of the sides AD, AE. Q.E.D.

LEMMA X.

The spaces which a bodij describes by any finite force urging it. whether

that force is determined and immutable, or is continually augmented

or continually diminished, are in the very beginning of the motion one

to the other in the duplicate ratio of the times.

Let the times be represented by the lines AD, AE, and the velocities
generated in those times by the ordinates DB, EC. The spaces described
with these velocities will be as the areas ABD, ACE. described by those
ordinates, that is, at the very beginning of the motion (by Lem. IX), in
the duplicate ratio of the times AD, AE. Q..E.D.

COR. 1. And hence one may easily infer, that the errors of bodies des
cribing similar parts of similar figures in proportional times, are nearly
as the squares of the times in which they are generated ; if so be these
errors are generated by any equal forces similarly applied to the bodies,
and measured by the distances of the bodies from those places of the sim
ilar figures, at which, without the action of those forces, the bodies would
have arrived in those proportional times.

COR. 2. But the errors that are generated by proportional forces, sim
ilarly applied to the bodies at similar parts of the similar figures, are as
the forces and the squares of the times conjuiu tly.

COR. 3. The same thing is to be understood of any spaces whatsoever
described by bodies urged with different forces ; all which, in the very be-
g'nning of the motion, are as the forces and the squares of the times conjunctly.

100

THE MATHEMATICAL PRINCIPLES

I SEC. 1

COR. 4. And therefore the forces are as the spaces described in the very
beginning of the motion directly, and the squares of the times inversely.

COR. 5. And the squares of the times are as the spaces described direct
ly, und the forces inversely.

SCHOLIUM.

If in comparing indetermined quantities of different sorts one with
another, any one is said to be as any other directly or inversely, the mean
ing is, that the former is augmented or diminished in the same ratio with
the latter, or with its reciprocal. And if any one is said to be as any other
two or more directly or inversely, the meaning is, that the first is aug
mented or diminished in the ratio compounded of the ratios in which the
others, or the reciprocals of the others, are augmented or diminished. As
if A is said to be as B directly, and C directly, and D inversely, the mean
ing is, that A is augmented or diminished in the same ratio with B X C
X -jj-, that is to say, that A and -£• arc one to the other in a given ratio.

LEMMA XL

The evanescent subtense of the angle of contact, in all curves which at
the point of contact have a finite curvature, is ultimately in the dupli
cate rati1) of the subtense of the conterminate arc.
CASE 1. Let AB be that arc, AD its tangent, BD
the subtense of the angle of contact perpendicular on
the tangent, AB the subtense of the arc. Draw BG
perpendicular to the subtense AB, and AG to the tan
gent AD, meeting in G ; then let the points D, B, and
G. approach to the points d, b, and g, and suppose J
to be the ultimate intersection of the lines BG, AG,
when the points D, B, have come to A. It is evident
that the distance GJ may be less than any assignable.
But (from the nature of the circles passing through
the points A, B, G, A, b, g,) AE2= AG X BD, and
A62 = Ag X bd ; and therefore the ratio of AB2 to Ab2 is compounded oi
the ratios of AG to Ag, and of Ed to bd. But because GJ may be as
sumed of less length than any assignable, the ratio of AG to Ag may be
such as to differ from the ratio of equality by less than any assignable
difference ; and therefore the ratio of AB2 to Ab2 may be such as to differ
from the ratio of BD to bd by less than any assignable difference. There
fore, by Lem. I, the ultimate ratio of AB2 to Ab2 is the same with tho ul
timate ratio of BD to bd. Q.E.D.

CASE 2. Now let BD be inclined to AD in any given an*r1r, and the
ultimate ratio of BD to bd will always be the same as before, and there
fore the same with the ratio of AB2 to Ab2. Q.E-P

BOOK I.]

OF NATURAL PHILOSOPHY.

101

CASE 3. And if we suppose the angle D not to be given, but that the
right line BD converges to a given point, or is determined by any other
condition whatever ; nevertheless the angles D, d, being determined by the
same law, will always draw nearer to equality, arid approach nearer to
each other than by any assigned difference, and therefore, by Lem. I, will at
lust be* equal ; and therefore the lines BD; bd arc in the same ratio to each
other as before. Q.E.D.

COR. 1. Therefore since the tangents AD, Ad, the arcs AB, Ab, and
their sines, BC, be, become ultimately equal to the chords AB, Ab} their
squares will ultimately become as the subtenses BD, bd.

COR. 2. Their squares are also ultimately as the versed sines of the arcs,
bisecting the chords, and converging to a given point. For those versed
sines are as the subtenses BD, bd.

COR. 3. And therefore the versed sine is in the duplicate ratio of the
time in which a body will describe the arc with a given velocity.

COR. 4. The rectilinear triangles ADB, Adb are
ultimately in the triplicate ratio of the sides AD, Ad, c
and in a sesquiplicate ratio of the sides DB, db ; as
being in the ratio compounded of the sides AD to DB,
and of Ad to db. So also the triangles ABC, Abe
are ultimately in the triplicate ratio of the sides BC, be.
What I call the sesquiplicate ratio is the subduplicate
of the triplicate, as being compounded of the simple
and subduplicate ratio. j

COR. 5. And because DB, db are ultimately paral- g
lei and in the duplicate ratio of the lines AD, Ad, the
ultimate curvilinear areas ADB, Adb will be (by the nature of the para
bola) two thirds of the rectilinear triangles ADB, Adb and the segments
AB, Ab will be one third of the same triangles. And thence those areas
and those segments will be in the triplicite ratio as well of the tangents
AD, Ad, as of the chords and arcs AB, AB.

SCHOLIUM.

But we have all along supposed the angle of contact to be neither infi
nitely greater nor infinitely less than the angles of contact made by cir
cles and their tangents ; that is, that the curvature at the point A is neither
infinitely small nor infinitely great, or that the interval AJ is of a finite mag
nitude. For DB may be taken as AD3 : in which case no circle can be drawn
through the point A, between the tangent AD and the curve AB, and
therefore the angle of contact will be infinitely less than those of circles.
And by a like reasoning, if DB be made successfully as AD4, AD5, AD8,
AD7, etc., we shall have a series of angles of contact, proceeding in itifini-
tum, wherein every succeeding term is infinitely less than the pre-

102 THE MATHEMATICAL PRINCIPLES [BOOK 1

ceding. And if DB be made successively as AD2, AD|, AD^, AD], AD|
AD7, &c., we shall have another infinite series of angles of contact, the first
of which is of the same sort with those of circles, the second infinitely
greater, and every succeeding one infinitely greater than the preceding.
But between any two of these angles another series of intermediate angles
of contact may be interposed, proceeding both ways in infinitum. wherein
every succeeding angle shall be infinitely greater or infinitely less than the
preceding. As if between the terms AD2 and AD3 there were interposed
the series AD'f, ADy, AD49, AD|, AD?, AD|, AD^1, AD^, AD^7, &c. And
again, between any two angles of this series, a new series of intermediate
angles may be interposed, differing from one another by infinite intervals.
Nor is nature confined to any bounds.

Those things which have been demonstrated of curve lines, and the
euperfices which they comprehend, may be easily applied to the curve su-
perfices and contents of solids. These Lemmas are premised to avoid the
tediousness of deducing perplexed demonstrations ad absurdnm, according
to the method of the ancient geometers. For demonstrations are more
contracted by the method of indivisibles : but because the hypothesis of
indivisibles seems somewhat harsh, and therefore that method is reckoned
less geometrical, I chose rather to reduce the demonstrations of the follow
ing propositions to the first and last sums and ratios of nascent and evane
scent quantities, that is, to the limits of those sums and ratios ; and so to
premise, as short as I could, the demonstrations of those limits. For hereby
the same thing is performed as by the method of indivisibles ; and now
those principles being demonstrated, we may use them with more safety.
Therefore if hereafter I should happen to consider quantities as made up of
particles, or should use little curve lines for right ones, I would not be un-
(lerstood to mean indivisibles, but evanescent divisible quantities : not the
sums and ratios of determinate parts, but always the limits of sums and
ratios ; and that the force of such demonstrations always depends on the
method laid down in the foregoing Lemmas.

Perhaps it may be objected, that there is no ultimate proportion, of
evanescent quantities ; because the proportion, before the quantities have
vanished, is not the ultimate, and when they are vanished, is none. But
by the same argument, it may be alledged, that a body arriving at a cer
tain place, and there stopping has no ultimate velocity : because the velo
city, before the body comes to the place, is not its ultimate velocity ; when
it has arrived, is none i ut the answer is easy; for by the ultimate ve
locity is meant that with which the body is moved, neither before it arrives
at its last place and the motion ceases, nor after, but at the very instant it
arrives ; that is, that velocity with which the body arrives at its last place,
and with which the motion ceases. And in like manner, by the ultimate ra
tio of evanescent quantities is to Le understood the ratio of the ijuantitiea

SEC. II.] OF NATURAL PHILOSOPHY. 103

not before they vanish, nor afterwards, but with which they vanish. In
like manner the first ratio of nascent quantities is that with which they begin
to be. And the first or last sum is that with which they begin and cease
to be (or to be augmented or diminished). There is a limit which the ve
locity at the end of the motion may attain, but not exceed. This is the
ultimate velocity. And there is the like limit in all quantities and pro
portions that begin and cease to be. And since such limits are certain and
definite, to determine the same is a problem strictly geometrical. But
whatever is geometrical we may be allowed to use in determining and de
monstrating any other thing that is likewise geometrical.

It may also be objected, that if the ultimate ratios of evanescent quan
tities are given, their ultimate magnitudes will be also given : and so all
quantities will consist of indivisibles, which is contrary to what Euclid
has demonstrated concerning incommensurables, in the 10th Book of his
Elements. But this objection is founded on a false supposition. For
those ultimate ratios with which quantities vanish are not truly the ratios
of ultimate quantities, but limits towards which the ratios of quantities
decreasing without limit do always converge ; and to which they approach
nearer than by any given difference, but never go beyond, nor in effect attain
to, till the quantities are diminished in wfinitum. This thing will appear
more evident in quantities infinitely great. If two quantities, whose dif
ference is given, be augmented in infin&um, the ultimate ratio of these
quantities will be given, to wit, the ratio of equality ; but it does not from
thence follow, that the ultimate or greatest quantities themselves, whose
ratio that is, will be given. Therefore if in what follows, for the sake of
being more easily understood, I should happen to mention quantities as
least, or evanescent, or ultimate, you are not to suppose that quantities of
any determinate magnitude are meant, but such as are conceived to be al
ways diminished without end.

SECTION II.

Of the Invention of Centripetal Forces.

PROPOSITION I. THEOREM 1.

The areas, which revolving bodies describe by radii drawn to an ^mmo-

vable centra of force do lie in tJ:e same immovable planes, and are pro-

portional to the times in which they are described.

For suppose the time to be divided into equal parts, and in the first part

of that time let the body by its innate force describe the right line AB

In the second part of that time, the same would (by Law I.), if not hindered,

proceel directly to c, alo ILJ; the line Be equal to AB ; so that by the radii

AS, BS, cS, draw. i to the centre, the equal areas ASB, BSc, would be de-

104

THE MATHEMATICAL PRINCIPLES

[BOOK I

scribed. But when the body
is arrived at B, suppose
that a centripetal force acts
at once with a great im
pulse, and, turning aside the
body from the right line Be,
compels it afterwards to con
tinue its motion along the
right line BC. Draw cC
parallel to BS meeting BC
in C ; and at the end of the
second part of the time, the
body (by Cor. I. of the Laws)
will be found in C, in the
same plane with the triangle
A SB. Join SC, and, because s
SB and Cc are parallel, the triangle SBC will be equal to the triangle SBc,
and therefore also to the triangle SAB. By the like argument, if the
centripetal force acts successively in C, D, E. &c., and makes the body, in
each single particle of time, to describe the right lines CD, DE, EF7 &c.,
they will all lie in the same plane : and the triangle SCD will be equal to
the triangle SBC, and SDE to SCD, and SEF to SDE. And therefore,
in equal times, equal areas are described in one immovable plane : and, by
composition, any sums SADS, SAFS, of those areas, are one to the other
as the times in which they are described. Now let the number of those
triangles be augmented, and their breadth diminished in wjinitum ; and
(by Cor. 4, Lem. III.) their ultimate perimeter ADF will be a curve line :
and therefore the centripetal force, by which the body is perpetually drawn
back from the tangent of this curve, will act continually ; and any described
areas SADS, SAFS, which are always proportional to the times of de
scription, will, in this case also, be proportional to those times. Q.E.D.

COR. 1. The velocity of a body attracted towards an immovable centre,
in spaces void of resistance, is reciprocally as the perpendicular let fall
from that centre on the right line that touches the orbit. For the veloci
ties in those places A, B, C, D, E. are as the bases AB, BC, CD, DE, EF.
of equal triangles ; and these bases are reciprocally as the perpendiculars
let fall upon them.

COR. 2. If the chords AB, BC of two arcs, successively described in
equal times by the same body, in spaces void of resistance, are completed
into a parallelogram ABCV, and the diagonal BV of this parallelogram;
in the position which it ultimately acquires when those arcs are diminished
in irifinitum, is produced both ways, it will pass through the centre of force.

COR. 3. If the chords AB, BC, and DE, EF, cf arcs described in equal

SEC. II.]

OF NATURAL PHILOSOPHY.

105

times, in spaces void of resistance, are completed into the parallelograms
ABCV, DEFZ : the forces in B and E are one to the other in the ulti
mate ratio of the diagonals BV, EZ, when those arcs are diminished in
infinitum. For the motions BC and EF of the body (by Cor. 1 of the
Laws) are compounded of the motions Be, BV, and E/", EZ : but BV and
EZ, which are equal to Cc and F/, in the demonstration of this Proposi
tion, were generated by the impulses of the centripetal force in B and E;
and are therefore proportional to those impulses.

COR. 4. The forces by which bodies, in spaces void of resistance, are
drawn back from rectilinear motions, and turned into curvilinear orbits,
are one to another as the versed sines of arcs described in equal times ; which
versed sines tend to the centre of force, and bisect the chords when those
arcs are diminished to infinity. For such versed sines are the halves of
the diagonals mentioned in Cor. 3.

COR. 5. And therefore those forces are to the force of gravity as the said
versed sines to the versed sines perpendicular to the horizon of those para
bolic arcs which projectiles describe in the same time.

COR. 6. And the same things do all hold good (by Cor. 5 of the Laws),
when the planes in which the bodies are moved, together with the centres
of force which are placed in those planes, are not at rest, but move uni
formly forward in right lines.

PROPOSITION II. THEOREM II.

Every body that moves in any curve line described in a plane, and by a
radius, drawn to a point either immovable, or moving forward with
an uniform rectilinear motion, describes about that point areas propor
tional to the times, is urged by a centripetal force directed to that point
CASE. 1. For every body

that moves in a curve line,

is (by Law 1) turned aside

from its rectilinear course

by the action of some force

that impels it. And that force

by which the body is turned

off from its rectilinear course,

and is made to describe, in

equal times, the equal least

triangles SAB, SBC, SCD,

&c., about the immovable

point S (by Prop. XL. Book

1, Elem. and Law II), acts

in the place B, according to

the direction of a line

par-

1U6 THE MATHEMATICAL PRINCIPLES [BOOK f.

allel K cC. that is, in the direction of the line BS. and in the place C,
accordii g to the direction of a line parallel to dD, that is, in the direction
of the line CS, (fee.; and therefore acts always in the direction of lines
tending to the immovable point S. Q.E.I).

CASE. 2. And (by Cor. 5 of the Laws) it is indifferent whether the su-
perfices in which a body describes a curvilinear figure be quiescent, or moves
together with the body, the figure described, and its point S, uniformly
forward in right lines.

COR. 1. In non-resisting spaces or mediums, if the areas are not propor
tional to the times, the forces are not directed to the point in which the
radii meet ; but deviate therefrom in. consequently or towards the parts to
which the motion is directed, if the description of the areas is accelerated ;
but in antecedentia, if retarded.

COR. 2. And even in resisting mediums, if the description of the areas
is accelerated, the directions of the forces deviate from the point in which
the radii meet, towards the parts to which the motion tends.

SCHOLIUM.

A body may be urged by a centripetal force compounded of several
forces ; in which case the meaning of the Proposition is, that the force
which results out of all tends to the point S. But if any force acts per
petually in the direction of lines perpendicular to the described surface,
this force will make the body to deviate from the plane of its motion : but
will neither augment nor diminish the quantity of the described surface
and is therefore to be neglected in the composition of forces.

PROPOSITION III. THEOREM III.

Every body, that by a radius drawn to the centre of another body, how
soever moved, describes areas about that centre proportional to iJie times,
is urged by a force compounded out of the centripetal force Bending fo
that other body, and of all the accelerative force by which that other
body is impelled.

Let L represent the one, and T the other body ; and (by Cor. 0 of the Laws)
if both bodies are urged in the direction of parallel lines, by a ne T force
equal and contrary to that by which the second body T is tinned, the first
body L will go on to describe about the other body T the same areas as
before : but the force by which that other body T was urged will be now
destroyed by an equal and contrary force; and therefore (by Law I.) that
other body T, now left to itself, will either rest, or move uniformly forward
in a right line : and the first body L impelled by the difference of the
forces, that is, by the force remaining, will go on to describe about the other
body T areas proportional to the times. And therefore (by Theor. II.) the
difference ;f the forces is directed to the other body T as its centre. Q.E.D

SEC. IL] OF NATURAL PHILOSOPHY. 107

Co.*. 1. Hence if the one body L, by a radius drawn to the other body T,
describes areas proportional to the times ; and from the whole force, by which
the firr.t body L is urged (whether that force is simple, or, according to
Cor. 2 of the Laws, compounded out of several forces), we subduct (by the
same Cor.) that whole accelerative force by which the other body is urged ;
the who_e remaining force by which the first body is urged will tend to the
( ther body T, as its centre.

COR. 2. And, if these areas are proportional to the times nearly, the re
maining force will tend to the other body T nearly.

COR. 3. And vice versa, if the remaining force tends nearly to the other
body T, those areas will be nearly proportional to the times.

COR. 4. If the body L, by a radius drawn to the other body T, describes
areas, which, compared with the times, are very unequal ; and that other
body T be either at rest, or moves uniformly forward in a right line : the
action of the centripetal force tending to that other body T is either none
at all, or it is mixed and compounded with very powerful actions of other
forces : and the whole force compounded of them all, if they are many, is
directed to another (immovable or moveaJble) centre. The same thing ob
tains, when the other body is moved by any motion whatsoever ; provided
that centripetal force is taken, wrhich remains after subducting that whole
force acting upon that other body T.

SCHOLIUM.

Because the equable description of areas indicates that a centre is re
spected by that force with which the body is most affected, and by which it
is drawn back from its rectilinear motion, and retained in its orbit ; why
may we not be allowed, in the following discourse, to use the equable de
scription of areas as an indication of a centre, about which all circular
motion is performed in free spaces ?

PROPOSITION IV. THEOREM IV.

The centripetal forces of bodies, which by equable 'motions describe differ
ent circles, tend to the centres of the same circles ; and are one to tJie
other as the squares of t/ie arcs described in equal times applied to the
radii of the circles.

These forces tend to the centres of the circles (by Prop. II., and Cor. 2,
Prop. L), and are one to another as the versed sines of the least arcs de
scribed in equal times (by Cor. 4, Prop. I.) ; that is, as the squares of the
same arcs applied to the diameters of the circles (by Lem. VII.) ; and there
fore since those arcs are as arcs described in any equal times, and the dia-
me ers ace as the radii, the forces will be as the squares of any arcs de-
scr bed in the same time applied to the radii of the circles. Q.E.D.
^OR. 1. Therefore, since those arcs are as the velocities of the bodies.

I OS THE MATHEMATICAL PRINCIPLES [BOOK .

the centripetal forces are in a ratio compounded of the duplicate ra'jio of
the velocities directly, and of the simple ratio of the radii inversely.

COR. 2. And since the periodic times are in a ratio compounded of the
ratio of the radii directly, and the ratio of the velocities inversely, the cen
tripetal forces, are in a ratio compounded of the ratio of the radii directly,
and the duplicate ratio of the periodic times inversely.

COR, 3. Whence if the periodic times are equal, and the velocities
therefore as the radii, the centripetal forces will be also as the radii ; and
tke contrary.

COR. 4. If the periodic times and the velocities are both in the subdu-
plicate ratio of the radii, the centripetal forces will be equal among them
selves ; and the contrary.

COR. 5. If the periodic times are as the radii, and therefore the veloci
ties equal, the centripetal forces will be reciprocally as the radii ; and the
contrary.

COR. 6. If the periodic times are in the sesquiplicate ratio of the radii,
and therefore the velocities reciprocally in the subduplicate ratio of the
radii, the centripetal forces will be in the duplicate ratio of the radii in
versely : and the contrary.

COR. 7. And universally, if the periodic time is as any power Rn of the
radius R, and therefore the velocity reciprocally as the power Rn ] of
the radius, the centripetal force will be reciprocally as the power R2n 1 of
the radius; and the contrary.

COR. 8. The same things all hold concerning the times, the velocities,
and forces by which bodies describe the similar parts of any similar figures
that have their centres in a similar position with those figures ; as appears
by applying the demonstration of the preceding cases to those. And the
application is easy, by only substituting the equable description of areas in
the place of equable motion, and using the distances of the bodies from the
centres instead of the radii.

COR. 9. From the same demonstration it likewise follows, that the arc
which a body, uniformly revolving in a circle by means of a given centri
petal force, describes in any time, is a mean proportional between the
diameter of the circle, and the space which the same body falling by the
same given force would descend through in the same given time.

SCHOLIUM.

The case of the 6th Corollary obtains in the celestial bodies (as Sir
Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed) ;
and therefore in what follows, I intend to treat more at large of those
things which relate to centripetal force decreasing in a duplicate ratio
of the distances from the centres.

Moreover, by means of the preceding Proposition and its Corollaries, we

SEC. II.] OF NATURAL PHILOSOPHY. 109

may discover the proportion of a centripetal force to any other known
force, such as that of gravity. For if a body by means of its gravity re
volves in a circle concentric to the earth, this gravity is the centripetal
force of that body. But from the descent of heavy bodies, the time of one
entire revolution, as well as the arc described in any given time, is given
(by Cor. 9 of this Prop.). And by such propositions, Mr. Huygens, in his
excellent book De Horologio Oscillatorio, has compared the force of
gravity with the centrifugal forces of revolving bodies.

The preceding Proposition may be likewise demonstrated after this
manner. In any circle suppose a polygon to be inscribed of any number
of sides. And if a body, moved with a given velocity along the sides of the
polygon, is reflected from the circle at the several angular points, the force,
with which at every reflection it strikes the circle, will be as its velocity :
and therefore the sum of the forces, in a given time, will be as that ve
locity and the number of reflections conjunctly ; that is (if the species of
the polygon be given), as the length described in that given time, and in
creased or diminished in the ratio of the same length to the radius of the
circle ; that is, as the square of that length applied to the radius ; and
therefore the polygon, by having its sides diminished in'inftnitum, coin
cides with the circle, as the square of the arc described in a given time ap
plied to the radius. This is the centrifugal force, with which the body
impels the circle ; and to which the contrary force, wherewith the circle
continually repels the body towards the centre, is equal.

PROPOSITION V. PROBLEM I.

There being given, in any places, the velocity with which a body de
scribes a given figure, by means of forces directed to some common
centre : to find that centre.

Let the three right lines PT, TQV, VR
touch the figure described in as many points,
P, Q, R, and meet in T and V. On the tan
gents erect the perpendiculars PA, QB, RC,
reciprocally proportional to the velocities of the
body in the points P, Q, R, from which the
perpendiculars were raised ; that is, so that PA
may be to QB as the velocity in Q to the velocity in P, and QB to RC
as the velocity in R to the velocity in Q. Through the ends A, B, C, of
the perpendiculars draw AD, DBE, EC, at right angles, meeting in D and
E : and the right lines TD, VE produced, will meet in S, the centre re
quired.

For the perpendiculars let fall from the centre S on the tangents PT.
QT. are reciprocally as the velocities of the bodies in the points P and Q

110 THE MATHEMATICAL PRINCIPLES [BOOK 1

(by Cor. 1, Prop. I.), and therefore, by construction, as the perpendiculars
AP, BQ, directly ; that is, as the perpendiculars let fall from the point D
on the tangents. Whence it is easy to infer that the points S, D, T, are
in one right line. And by the like argument the points S, E, V are also
in one right line ; and therefore the centre S is in the point where the
right lines TD; YE meet. Q.E.D.

PROPOSITION VL THEOREM V.

In a space void of resistance, if a body revolves in any orbit about an im
movable centre, and in the least time describes any arc just then, na
scent ; and the versed sine of that arc is supposed to be drawn bisect
ing the chord, and produced passing through the centre of force: the
centripetal force in the middle of the arc will be as the versed sine di
rectly and the square of the time inversely.
For the versed sine in a given time is as the force (by Cor. 4, Prop. 1) ;

and augmenting the time in any ratio, because the arc will be augmented

in the same ratio, the versed sine will be augmented in the duplicate of

that ratio (by Cor. 2 and 3, Lem. XL), and therefore is as the force and the

square of the time. Subduct on both sides the duplicate ratio of the

time, and the force will be as the versed sine directly, arid the square of

the time inversely. Q.E.D.

And the same thing may also be easily demonstrated by Corol. 4?

T,em. X.

COR. 1. If a body P revolving about the

centre S describes a curve line APQ,, which a

right line ZPR touches in any point P ; and

from any other point Q, of the curve, QJl is

drawn parallel to the distance SP, meeting

the tangent in R ; and QT is drawn perpen-

(licular to the distance SP ; the centripetal force will be reciprocally as the

sp2 x Q/r2

solid - — :— , if the solid be taken of that magnitude which it ulti-

mately acquires when the points P and Q, coincide. For Q,R is equal to
the versed sine of double the arc QP, whose middle is P : and double the
triangle SQP, or SP X Q,T is proportional to the time in which that
double arc is described ; and therefore may be used for the exponent of
the time.

COR. 2. By a like reasoning, the centripetal force is reciprocally as the

SY2 X QJP2
solid - 7^5 - ; if SY is a perpendicular from the centre of force on

PR the tangent of the orbit. For the rectangles SY X QP and SP X Q,T
are equal.

SEC. II.] OF NATURAL PHILOSOPHY. Ill

COR. 3. If the orbit is cither a circle, or touches or cuts a circle c< ncen-
trically, that is, contains with a circle the least angle of contact or sec
tion, having the same curvature rnd the same radius of curvature at the
point P : and if PV be a chord of this circle, drawn from the body through
the centre of force ; the centripetal force will be reciprocally as the solid

QP2

SY2 X PV. For PV is - .

COR. 4. The same things being supposed, the centripetal force is as the
square of the velocity directly, and that chord inversely. For the velocity
is reciprocally as the perpendicular SY, by Cor. 1. Prop. I.

COR. 5. Hence if any curvilinear figure APQ, is given, and therein a
point S is also given, to which a centripetal force is perpetually directed.
that law of centripetal force may be found, by which the body P will bcj
continually drawn back from a rectilinear course, and. being detained in
the perimeter of that figure, will describe the same by a perpetual revolu-

SP2 x QT2

tion. That is, we are to find, by computation, either the solid ---- -

or the solid SY2 X PV, reciprocally proportional to this force. Example:
of this we shall give in the following Problems.

PROPOSITION VII. PROBLEM II.

Tf a body revolves in the circumference of a circle; it is proposed to finii
the law of centripetal force directed to any given, point.

Let VQPA be the circumference of the
circle ; S the given point to which as to
a centre the force tends : P the body mov
ing in the circumference ; Q the next
place into which it is to move; and PRZ
the tangent of the circle at the preceding
place. Through the point S draw the v
chord PV, and the diameter VA of the
circle : join AP, and draw Q,T perpen
dicular to SP, which produced, may meet
the tangent PR in Z ; and lastly, through
the point Q, draw LR parallel to SP, meeting the circle" in L, and the
tangent PZ in R. And, because of the similar triangles ZQR, ZTP.
VPA, we shall have RP2, that is. QRL to QT2 as AV2 to PV2. And

QRlj x PV2 SI3-'

therefore '- —TS -- is equal to QT2. Multiply those equals by -'.

and the points P and Q, coinciding, for RL write PV ; then we shall have

SP-' X PV5 SP2 x QT2

— • And therefore fl»r Cor 1 and 5. Prop. VI.)

112 THE MATHEMATICAL PRINCIPLES [BOOK I,

SP2 X PV3

the centripetal force is reciprocally as - — ry^~ — J that is (because AV2

ia given), reciprocally as the square of the distance or altitude SP, and the
3ube of the chord PV conjunctly. Q.E.L

The same otherwise.

On the tangent PR produced let fall the perpendicular SY ; and (be
cause of the similar triangles SYP, VPA), we shall have AV to PV as SP

SP X PV SP2 >< PV3

to SY, and therefore -- ^~— - = SY, and - — ^- = SY2 X PV.
A V A V

And therefore (by Corol. 3 and 5, Prop. VI), the centripetal force is recip-

SP2 X PV3

rocally as - ~~ry¥~~~ I *na* *s (because AV is given), reciprocally as SP"

X PV3. Q.E.I.

Con. 1. Hence if the given point S, to which the centripetal force al
ways tends, is placed in the circumference of the circle, as at V, the cen
tripetal force will be reciprocally as the quadrato-cube (or fifth power) of
the altitude SP.

COR. 2. The force by which the body P in the
circle APTV revolves about the centre of force S
is to the force by which the same body P may re
volve in the same circle, and in the same periodic
time, about any other centre of force R, as RP2 X
SP to the cube of the right line SG, which, from
the first centre of force S is drawn parallel to the
distance PR of the body from the second centre of force R, meeting the
tangent PG of the orbit in G. For by the construction of this Proposition,
the former force is to the latter as RP2 X PT3 to SP2 X PV3; that is, as

SP3 X PV3

SP X RP2 to -- p™ — ; or (because of the similar triangles PSG, TPV)

to SGS.

COR. 3. The force by which the body P in any orbit revolves about the
centre of force S, is to the force by which the same body may revolve in
the same orbit, and the same periodic time, about any other centre of force
R. as the solid SP X RP2, contained under the distance of the body from
the first centre of force S, and the square of its distance from the sec
ond centre of force R, to the cube of the right line SG, drawn from the
first centre of the force S, parallel to the distance RP of the body from
fch*3 second centre of force R, meeting the tangent PG of the orbit in G.
For the force in this orbit at any point P is the same as in a circle of the
same curvature.

SJSG. IL]

OF NATURAL PHILOSOPHY.

113

PROPOSITION VIII. PROBLEM III.

If a body mi ues in the semi-circuwferencePQA: it is proposed to find
the law of the centripetal force tending to a point S, so remote, that all
the lines PS. RS drawn thereto, may be taken for parallels.
From C, the centre of the semi-circle, let

the semi-diameter CA he drawn, cutting the

parallels at right angles in M and N, and

join CP. Because of the similar triangles

CPM, PZT, and RZQ, we shall have CP2

to PM2 as PR2 to QT2; and, from the na

ture of the circle, PR2 is equal to the rect

angle QR X RN + QN, or, the points P, Q coinciding, to the rectangle

QR x 2PM. Therefore CP2 is to PM2 as QR X 2PM to QT2; and

QT2 2PM3 QT2 X SP2 2PM3 X SP2

therefore (by

QR

Corol.
8PM3 X SP2

,
and

QR

And

1 and 5, Prop. VI.), the centripetal force is reciprocally as

2SP2

.
that is (neglecting the given ratio -ppr)> reciprocally as

PM3. Q.E.L

And the same thing is likewise easily inferred from the preceding Pro
position.

SCHOLIUM.

And by a like reasoning, a body will be moved in an ellipsis, or even ia
an hyperbola, or parabola, by a centripetal force which is reciprocally ae
the cube of the ordinate directed to an infinitely remote centre of force.

PROPOSITION IX. PROBLEM IV.

If a body revolves in a spiral PQS, cutting all the radii SP, SQ, fyc.,
in a given angle; it is proposed to find thelaio of the centripetal force
tending to tJie centre of that spiral.
Suppose the inde

finitely small angle AY

PSQ to be given ; be

cause, then, all the

angles are given, the

figure SPRQT will , _

be given in specie. v

QT Q,T2

Therefore the ratio -7^- is also given, and „„ is as QT, that is (be
lot IX QK

cause the figure is given in specie), as SP. But if the angle PSQ is any
way changed, the right line QR, subtending the angle of contact QPU

tU

THE MATHEMATICAL PRINCIPLES

[BOOK J

(by Lemma XI) will be changed in the duplicate ratio of PR or QT

QT2

Therefore the ratio ~TVD~ remains the same as before, that is, as SP. And

QT2 x SP2

— -^ is as SP3, and therefore (by Corol. 1 and 5, Prop. YI) the

centripetal force is reciprocally as the cube of the distance SP. Q.E.I.

The same otherwise.

The perpendicular SY let fall upon the tangent, and the chord PY of
the circle concentrically cutting the spiral, are in given ratios to the height
SP ; and therefore SP3 is as SY2 X PY, that is (by Corol. 3 and 5, Prop.
YI) reciprocally as the centripetal force.

LEMMA XII.

All parallelograms circumscribed about any conjugate diameters of a
given ellipsis or hyperbola are equal among themselves.
This is demonstrated by the writers on the conic sections.

PROPOSITION X. PROBLEM Y.

If a body revolves in an ellipsis ; it is proposed to find the law of thi
centripetal force tending to the centre of the ellipsis.

Suppose CA, CB to
be semi-axes of the
ellipsis; GP, DK, con
jugate diameters ; PF,
Q,T perpendiculars to
those diameters; Qvan
^rdinate to the diame
ter GP ; and if the
parallelogram QvPR
be completed, then (by
the properties of the
jonic sections) the rec-
langle PvG will be to
Qv2 as PC2 to CD2;
and (because of the
similar triangles Q^T, PCF), Qi>2 to QT2 as PC2 to PF2 ; and, by com
position, the ratio of PvG to QT2 is compounded of the ratio of PC2 1<

QT2
CD2, and of the ratio of PC2 to PF2, that is, vG to -p— as PC;

to_92L^_P_]^_. Put QR for Pr, and (by Lem. XII) BC X CA for CD

K PF ; also (the points P and Q coinciding) 2PC for rG; and multiply-

SEC. II.] OF NATURAL PHILOSOPHY. 115

QT2 x PC2

ing the extremes and means together, we shall have rfo~ equal to

2BC2 X CA2

— pp — — . Therefore (by Cor. 5, Prop. VI), the centripetal force is

2BC2 X CA2

reciprocally as — — ry~ ; that is (because 2I3C2 X CA2 is given), re
ciprocally as-r^v; that is, directly as the distance PC. QEI.
I O

TJie same otherwise.

[n the right line PG on the other side of the point T, take the point u
so that Tu may be equal to TV ; then take uV, such as shall be to v G as
DC2 to PC2. And because Qr9 is to PvG as DC'2 to PC2 (by the conic
sections), we shall have Qv2 •-= Pi' X «V. Add the rectangle n.Pv to both
sides, and the square of the chord of the arc PQ, will be equal to the rect
angle VPv ; and therefore a circle which touches the conic section in P,
and passes through the point Q,, will pass also through the point V. Now
let the points P and Q, meet, and the ratio of nV to rG, which is the same
with the ratio of DC2 to PC2, will become the ratio of PV to PG, or PV

2DC2

to 2PC : and therefore PY will be equal to „„ — . And therefore the

force by which the body P revolves in the ellipsis will be reciprocally as

2 DC2

— ry— X PF2 (by Cor. 3, Prop. VI) ; that is (because 2DC2 X PF2 is
I O

given) directly as PC. Q.E.I.

COR. 1. And therefore the force is as the distance of the body from the
centre of the ellipsis ; and, vice versa, if the force is as the distance, the
body will move in an ellipsis whose centre coincides with the centre of force,
or perhaps in a circle into which the ellipsis may degenerate.

COR. 2. And the periodic times of the revolutions made in all ellipses
whatsoever about the same centre will be equal. For those times in sim
ilar ellipses will be equal (by Corol. 3 and S, Prop. IV) ; but in ellipses
that have their greater axis common, they are one to another as the whole
areas of the ellipses directly, and the parts of the areas described in the
same time inversely: that is, as the lesser axes directly, and the velocities
of the bodies in their principal vertices inversely ; :hat is, as those lesser
axes dirtily, and the ordinates to the same point %f the common axes in
versely ; and therefore (because of the equality of the direct and inverse
ratios) in the ratio of equality.

SCHOLIUM.

If the ellipsis, by having its centre removed to an infinite distance, de
generates into a parabola, the body will move in tin's parabola ; and the

116

THE MATHEMATICAL PRINCIPLES

[BOOK I

force, now tending to a centre infinitely remote, will become equable.
Which is Galileo s theorem. And if the parabolic section of the cone (by
changing the inclination of the cutting plane to the cone) degenerates into
an hyperbola, the body will move in the perimeter of this hyperbola, hav
ing its centripetal force changed into a centrifugal force. And in like
manner as in the circle, or in the ellipsis, if the forces are directed to the
centre of the figure placed in the abscissa, those forces by increasing or di
minishing the ordinates in any given ratio, or even by changing the angle
of the inclination of the ordinates to the abscissa, are always augmented
or diminished in the ratio of the distances from the centre ; provided the
periodic times remain equal ; so also in all figures whatsoever, if the ordi-
nates are augmented or diminished in any given ratio, or their inclination
is any way changed, the periodic time remaining the same, the forces di
rected to any centre placed in the abscissa are in the several ordinatee
augmented or diminished in the ratio of the distances from the centre

SECTION III.

Of the motion of bodies in eccentric conic sections.

PROPOSITION XL PROBLEM VI.

If a body revolves in an ellipsis ; it is required to find the law of the

centripetal force tending to the focus of the ellipsis.

Let S be the focus
of the ellipsis. Draw
SP cutting the diame
ter DK of the ellipsis
in E, and the ordinate
Qv in x ; and com
plete the parallelogram
d.rPR, It is evident
that EP is equal to the
greater semi-axis AC :
for drawing HI froln
the other focus H of
the ellipsis parallel to
EC, because CS, CH
are equal, ES, El will
be also equal ; so that EP is the half sum of PS, PI, that is (because of
the parallels HI, PR, and the equal angles IPR, HPZ), of PS, PH, which
taken together are equal to the whole axis 2AC. Draw QT perpendicu
lar to SP, and putting L for the princi al latus rectum of the ellipsis (or for

III. OF NATURAL PHILOSOPHY. 117

L X ^R t0 L X Py aS ^R t0 PV that 1S> US PE

or AC to PC ; and L X Pv to GvP as L to Gy ; and GvP to Qi>2 as
to CD- ; and by (Corol. 2, Lem. VII) the points Q, and P coinciding, Qv*
is to Q,r- in the ratio of equality ; and Q,.r2 or Qv2 is to Q,T2 as EP2 to
PF2, that is, as CA2 to PF2, or (by Lem. XII) as CD'2 to CB2. And com
pounding all those ratios together, we shall have L X QR to Q,T2 as AC
X L X PC2 X CD2, or 2CB2 X PC2 X CD2 to PC X Gv X CD2 X
CB2, or as 2PC to Gv. But the points Q and P coinciding, 2PC and Gr
are equal. And therefore the quantities L X QR and Q,T2, proportional

SP2

to these, will be also equal. Let those equals be drawn into-p^B"? and L

SP2 X QT2

X SP2 will become equal to -- ^p — — . And therefore (by Corol. 1 and

5, Prop. VI) the centripetal force is reciprocally as L X SP2, that is, re
ciprocally in the duplicate ratio of the distance SP. Q.E.I.

The same otherwise.

Since the force tending to the centre of the ellipsis, by which the body
P may revolve in that ellipsis, is (by Corol. 1, Prop. X.) as the distance
CP of the body from the centre C of the ellipsis ; let CE be drawn paral
lel to the tangent PR of the ellipsis : and the force by which the same body
P may revolve about any other point S of the ellipsis, if CE and PS in-

PE3

tersect in E, will be as ^T3, (by Cor. 3, Prop. VII.) ; that is, if the point

S is the focus of the ellipsis, and therefore PE be given as SP2 recipro
cally. Q.E.I.

With the same brevity with which we reduced the fifth Problem to the
parabola, and hyperbola, we might do the like here : but because of the
dignity of the Problem and its use in what follows, I shall confirm the other
cases by particular demonstrations.

PROPOSITION XII. PROBLEM VII.

Suppose a body to move in an hyperbola ; it is required to find lite law of

the centripetal force tending to the focus of that figure.

Let CA, CB be the semi-axes of the hyperbola ; PG, KD other con

jugate diameters ; PF a perpendicular to the diameter KD ; and Qv an

ordinate to the diameter GP. Draw SP cutting the diameter DK in E,

and the ordinate Qv in x, and complete the parallelogram QRP.r. It is

evident that EP is equal to the semi-transverse axis AC ; for drawing

HE, from the other focus H of the hyperbola, parallel to EC, because CS,

TH are equal, ES El will be also equal ; so that EP is the half difference

J1S

THE MATHEMATICAL PRINCIPLES

[Book I

.of PS, PI; that is (be
cause of the parallels IH,
PR, and the equal angles
IPR, HPZ), of PS, PH,
the difference of which is
equal to the whole axis
2AC. Draw Q,T perpen
dicular to SP; and put
ting L for the principal
latus rectum of the hy
perbola (that is, for

2BC2\ ....

-Tp- ) 7 we shall have L

X QR to L X Pv as QR
to Pv, or Px to Pv, that is
(because of the similar tri
angles Pxv, PEC), as PE
to PC, or AC to PC.
And L X Pv will be to
Gv X Pv as L to Gv;
and (by the properties of
the conic sections) the rec
tangle G?'P is to Q,v2 as
PC2 to CD2 ; and by (Cor. 2, Lem. VII.), Qv2 to Qa* the points Q and P
coinciding, becomes a ratio of equality ; and Q,.r2 or Qv2 is to Q,T2 as EP2
to PF2, that is, as CA2 to PF2, or (by Lem. XII.) as CD2 to CB2 : and,
compounding all those ratios together, we shall have L X Q,R to Q,T2 as
AC X L X PC2 X CD2, or 2CB2 X PC2 X CD2 to PC X Gv X CD2
X CB2, or as 2PC to Gv. But the points P and Q, coinciding. 2PC and
Gv are equal. And therefore the quantities L X Q,R arid Q.T2, propor
tional to them, will be also equal. Let those equals be drawn into

SP2 sp2 x o/r2

^, and we shall have L X SP2 equal to ^^ . And therefore (by

Cor. 1. and 5, Prop. VI.) the centripetal force is reciprocally as L X SP'2.
'hat is, reciprocally in the duplicate ratio of the distance SP. Q,.E.I.

TJie same otherwise.

Find out the force tending from the centre C of the hype rbola. This will
be proportional to the distance CP. But from thence (by Cor. 3, Prop.

PE3
VII.) the force tending to the focus S will be as -^-^ th; (t is, because PE

is given reciprocally as SP-. Q,.E.I.

SEC. III.]

OF NATURAL PHILOSOPHY.

119

And the same way may it be demonstrated, that the body having its cen
tripetal changed into a centrifugal force, will move in the conjugate hy
perbola.

LEMMA XIII.

The latus rectum of a parabola belonging to any vertex is quadruple
the distance of that vertex from the focus of thejigurc.

This is demonstrated by the writers on the conic sections.

LEMMA XIV.

Tlie perpendicular, let fall from the focus of a parabola on its tangent, is
a mean proportional between the. distances of the focus from the poini
of contact, and from the principal vertex of the figure.

For, let AP be the parabola, S its
focus, A its principal vertex, P the
point of contact, PO an ordinate to the
principal diameter. PM the tangent
meeting the principal diameter in M.

and SN the perpendicular from the fo-~ M A s o

cus on the tangent : join AN, and because of the equal lines MS and SP,
MN and NP, MA and AC, the right lines AN, OP, will be parallel ; and
thence the triangle SAN will be right-angled at A, and similar to the
equal triangles SNM, SNP j therefore PS is to SN as SN to SA. Q.E.D.

COR. 1. PS2 is to SN2 as PS to SA.

COR. 2. And because SA is given, SN- will be as PS.

COR. 3. And the concourse of any tangent PM, with the right line SN.
drawn from the focus per] endicular on the tangent, falls in the right line
AN that touches the parabola in the principal vertex.

PROPOSITION XIII. PROBLEM VIII.

If a body moves in the perimeter of a parabola ; it is required to find the.

law of the centripetal force tending to the focus of that figure.

Retaining the construction
of the preceding Lemma, let P
be the body in the perimeter
of the parabola ; and from the
place Q,, into which it is next
to succeed, draw QH parallel IS!.

and Q,T perpendicular to SP,
as also Qv parallel to the tan
gent, and mating the diame
ter PG in v, and the distance —

120 THE MATHEMATICAL PRINCIPLES [BOOK I.

SP in x. Now. because of the similar triangles Pxv, SPM, and of the
equal sides SP, SM of the one, the sides Px or Q,R and Pv of the other
will be also equal. But (by the conic sections) the square of the ordinate
Q,y is equal to the rectangle under the latus rectum and the segment Pv
of the diameter ; that is (by Lem. XIII.), to the rectangle 4PS X Pv, or
4PS X Q,R ; and the points P and Q, coinciding, the ratio of Qv to Q,.r
(by Cor. 2, Lem. VII.,) becomes a ratio of equality. And therefore Q,#2, in
this case, becomes equal to the rectangle 4PS X Q,R. But (because of the
similar triangles Q#T, SPN), Q^'2 is to QT2 as PS2 to SN2, that is (by
Cor. 1, Lem. XIV.), as PS to SA ; that is, as 4PS X QR to 4SA x QR,
and therefore (by Prop. IX. Lib. V., Elem.) QT* and 4SA X QR are

SP2 SP2 X QT2

equal. Multiply these equals by ^-^-, and ^5 — -will become equal

to SP2 X 4SA : and therefore (by Cor. 1 and 5, Prop. VL), the centripetal
force is reciprocally as SP2 X 4S A ; that is, because 4SA is given, recipro
cally in the duplicate ratio of the distance SP. Q.E.I.

COR. 1. From the three last Propositions it follows, that if any body P
goes from the place P with any velocity in the direction of any right line
PR, and at the same time is urged by the action of a centripetal force that
is reciprocally proportional to the square of the distance of the places from
the centre, the body will move in one of the conic sections, having its fo
cus in the centre of force ; and the contrary. For the focus, the point of
contact, and the position of the tangent, being given, a conic section may
be described, which at that point shall have a given curvature. But the
curvature is given from the centripetal force and velocity of the body be
ing given ; and two orbits, mutually touching one the other, cannot be de
scribed by the same centripetal force and the same velocity.

COR. 2. If the velocity with which the body goes from its place P is
such, that in any infinitely small moment of time the lineola PR may be
thereby describe I: and the centripetal force such as in the same time to
move the same body through the space QR ; the body will move in one of

QT2.

the conic sections, whose principal latus rectum is the quantity Tjfr in its

ultimate state, when thelineoke PR, QR are diminished in infinitum. In
these Corollaries I consider the circle as an ellipsis ; and I except the case
where the body descends to the centre in a right line.

PROPOSITION XIV. THEOREM VI.

Tf several bodies revolve about one common centre, and the centripetal
force is reciprocally in tlie duplicate ratio of the distance of places
from the centre ; I say, that the principal latera recta of tfieir orbits
are in the duplicate ratio of the areas, which the bodies by radii drawn
to the centre describe it\ the same time.

SEC. HI. OF NATURAL PHILOSO1 HY.

For (by Cor 2, Prop. XIII) the latus rectum

QT*.

L is equal to the quantity-^-in its ultimate

state when the points P and Q, coincide. But
the lineola QR in a given time is as the gen
erating centripetal force ; that is (by supposi-

QT2

tion), reciprocally as SP2. And therefore-^-^

is as Q.T2 X SP2 ; that is, the latus rectum L is in the duplicate ratio of
the area QT X SP. Q.E.D.

C?OR. Hence the whole area of the ellipsis, and the rectangle under the
axes, which is proportional to it, is in the ratio compounded of the subdu-
plicate ratio of the latus rectum, and the ratio of the periodic time. For
the whole area is as the area QT X SP, described in a given time, mul
tiplied by the periodic time.

PROPOSITION XV. THEOREM VII.

The same things being supposed, J say, that the periodic times in ellip
ses are in the sesquiplicate ratio of their greater axes.
For the lesser axis is a mean proportional between the greater axis and
the latus rectum ; and, therefore, the rectangle under the axes is in the
ratio compounded of the subduplicate ratio of the latus rectum and the
sesquiplicate ratio of the greater axis. But this rectangle (by Cor. o.
Prop. XIV) is in a ratio compounded of the subduplicate ratio of the
latus rectum, and the ratio of the periodic time. Subduct from both sides
the subduplicate ratio of the latus rectum, and there will remain the ses
quiplicate ratio of the greater axis, equal to the ratio of the periodic time.
Q.E.D.

COR. Therefore the periodic times in ellipses are the same as in circles
whose diameters are equal to the greater axes of the ellipses.

PROPOSITION XVI. THEOREM VIII.

The same things being supposed, and right lines being drawn to the
bodies that shall touch the orbits, and perpendiculars being let fall on
those tangents from the common focus ; I say, that the velocities oj
the bodies are in a ratio compounded of the ratio of the perpendiculars
inversely, and the, subduplicate ratio of the principal later a recta
direct!]).
From the focus S draw SY perpendicular to the tangent PR, and the

velocity of the body P will be reciprocally in the subduplicate ratio of the

SY2
quantity -y— . For that velocity is as the infinitely small arc PQ de-

122 THE MATHEMATICAL PRINCIPLES [BOOK I.

scribed in a given moment of time, that is (by
Lem. "VII), as the tangent PR ; that is (because
of the proportionals PR to Q,T, and SP to

SP X Q,T

SY), as ~y — — ; or as SY reciprocally,

and SP X Q,T directly ; but SP X QT is as
the area described in the given time, that is (by
Prop. XIV), in the subduplicate ratio of the
latus rectum. Q.E.D.

COR. 1. The principal latera recta are in a ratio compounded of the
duplicate ratio of the perpendiculars and the duplicate ratio of the ve
locities.

COR. 2. The velocities of bodies, in their greatest and least distances from
the common focus, are in the ratio compounded of the ratio of the distan
ces inversely, and the subduplicate ratio of the principal latera recta di
rectly. For those perpendiculars are now the distances.

COR. 3. Arid therefore the velocity in a conic section, at its greatest or
least distance from the focus, is to the velocity in a circle, at the same dis
tance from the centre, in the subduplicate ratio of the principal latus rec
tum to the double of that distance.

COR. 4. The velocities of the bodies revolving in ellipses, at their mean
distances from the common focus, are the same as those of bodies revolving
in circles, at the same distances ; that is (by Cor. 6. Prop. IV), recipro
cally in the subduplicate ratio of the distances. For the perpendiculars
are now the lesser semi-axes, and these are as mean proportionals between
the distances and the latera recta. Let this ratio inversely be compounded
with the subduplicate ratio of the latera recta directly, and we shall have
the subduplicate ratio of the distance inversely.

COR. 5. In the same figure, or even in different figures, whose principal
latera recta are equal, the velocity of a body is reciprocally as the perpen
dicular let fall from the focus on the tangent.

COR. 6. In a parabola, the velocity is reciprocally in the subduplicate
ratio.of the distance of the body from the focus of the figure; it is more
variable in the ellipsis, and less in the hyperbola, than according to this
ratio. For (by Cor. 2, Lem. XIV) the perpendicular let fall from the
focus on the tangent of a parabola is in the subduplicate ratio of the dis
tance. In the hyperbola the perpendicular is less variable ; in the ellipsis
more.

COR. 7. In a parabola, the velocity of a body at any distance from the
focus is to the velocity of a body revolving in a circle, at the same distance
from the centre, in the subduplicate ratio of the number 2 to 1 ; in the
ellipsis it is less, and in the hyperbola greater, than according to this ratio,
For (by Cor. 2 of this Prop.) the velocitv at the vertex of a parabola is ir

SEC. III.]

OF NATURAL PHILOSOPHY.

123

this ratio, and (by Cor. 6 of this Prop, and Prop. IV) the same proportion
holds in all distances. And hence, also, in a parabola, the velocity is
everywhere equal to the velocity of a body revolving in a circle at half the
distance ; in the ellipsis it is less, and in the hyperbola greater.

COR. S. The velocity of a body revolving in any conic section is to the
velocity of a body revolving in a circle, at the distance of half the princi
pal latus rectum of the section, as that distance to the perpendicular let
fall from the focus on the tangent of the section. This appears from
Cor. 5.

COR. 9. Wherefore since (by Cor. 6, Prop. IV), the velocity of a body
revolving in this circle is to the velocity of another body revolving in any
other circle reciprocally in the subduplicate ratio of the distances; there
fore, ex czqiiO) the velocity of a body revolving *in a conic section will be
to the velocity of a body revolving* in a circle at the same distance as a
mean proportional between that common distance, and half the principal
latus rectum of the section, to the perpendicular let fall from the common
focus upon the tangent of the section.

PROPOSITION XVII. PROBLEM IX.

Supposing the centripetal force to be reciprocally proportional to the
squares of the distances of places from the centre, and that the abso
lute quantity of that force is known ; it is required to determine t/te
line which a body will describe that is let go from a given place with a
given velocity in the direction of a given right line.
Let the centripetal force
tending to the point S be
such as will make the body
p revolve in any given orbit
pq ; and suppose the velocity
of this body in the place p
is known. Then from the
place P suppose the body P
to be let go with a given ve
locity in the direction of the
line PR ; but by virtue of a
centripetal force to be immediately turned aside from that right line into
the conic section PQ,. This, the right line PR will therefore touch in P.
Suppose likewise that the right line pr touches the orbit pq in p ; and if
from S you suppose perpendiculars let fall on those tangents, the principal
latus rectum of the conic section (by Cor. 1, Prop. XVI) will be to the
principal latus rectum of that orbit in a ratio compounded of the duplicate
ratio of the perpendiculars, and the duplicate ratio of the velocities ; arid
is therefore given. Let this latus rectum be L ; the focus S of the conic

L24 THE MATHEMATICAL PRINCIPLES [BOOK I

section is also given. Let the angle RPH be the complement of the angle
RPS to two right ; and the line PH, in which the other focus II is placed,
is given by position. Let fall SK perpendicular on PH, and erect the
conjugate semi-axis BC ; this done, we shall have SP2— 2KPH + PH2
= SH2 = 4CH2 = 4BH2 — 4BC2 = SP + PH2— L X SiM

SP2 + 2SPH + PH2 — L x SP + PH. Add on both sides 2KPH —

SP2— PH2 + L X SP + PH, and we shall have L X SP + PH = 2SPH
f 2KPH, or SP + PH to PH, as 2SP + 2KP to L. Whence PH is
given both in length and position. That is, if the velocity of the body
in P is such that the latus rectum L is less than 2SP + 2KP, PH will
lie on the same side of the tangent PR writh the line SP ; and therefore
the figure will be an ellipsis, which from the given foci S, H, and the
principal axis SP + PH, is given also. But if the velocity of the body
is so great, that the latus rectum L becomes equal to 2SP + 2KP, the
length PH will be infinite ; and therefore, the figure will be a parabola,
which has its axis SH parallel to the line PK, and is thence given. But
if the body goes from its place P with a yet greater velocity, the length
PH is to be taken on the other side the tangent ; and so the tangent pas
sing between the foci, the figure will be an hyperbola having its principal
axis equal to the difference of the lines SP and PH, and thence is given.
Por if the body, in these cases, revolves in a conic section so found, it is
demonstrated in Prop. XI, XII, and XIII, that the centripetal force will
be reciprocally as the square of the distance of the body from the centre
of force S ; and therefore we have rightly determined the line PQ,, which
a body let go from a given place P with a given velocity, and in the di
rection of the right line PR given by position, would describe with such a
force. Q.E.F.

COR. 1. Hence in every conic section, from the principal vertex D, the
latus rectum L, and the focus S given, the other focus H is given, by
taking DH to DS as the latus rectum to the difference between the latus
rectum and 4US. For the proportion, SP + PH to PH as 2SP + 2KP
to L, becomes, in the case of this Corollary, DS + DH to DH as 4DS to
L, and by division DS to DH as 4DS — L to L.

COR. 2. Whence if the velocity of a body in the principal vertex D ig
given, the orbit may be readily found ; to wit, by taking its latus rectum
to twice the distance DS, in the duplicate ratio of this given velocity to
the velocity of a body revolving in a circle at the distance DS (by Cor.
3, Prop. XVI.), and then taking DH to DS as the latus rectum to the
difference between the latus rectum and 4DS.

COR. 3. Hence also if a body move in any conic section, and is forced
out of its orbit by any impulse, you may discover the orbit in which it will
afterwards pursue its Bourse. For bv compounding the proper motion oi

SEC. IV.] OF NATURAL PHILOSOPHY. 125

the body with that motion, which the impulse alone would generate, you
will have the motion with which the body will go off from a given place
of impulse in the direction of a right line given in position.

COR. 4. And if that body is continually disturbed by the action of some
foreign force, we may nearly know its course, by collecting the changes
which that force introduces in some points, and estimating the continual
changes it will undergo in the intermediate places, from the analogy that
appears in the progress of the series.

SCHOLIUM.

If a body P, by means of a centripetal
force tending to any given point R, move
in the perimeter of any given conic sec
tion whose centre is C ; and the law of
the centripetal force is required : draw
CG parallel to the radius RP, and meet
ing the tangent PG of the orbit in G ;
and the force required (by Cor. 1, and

CG3
Schol. Prop. X., and Cor. 3, Prop. VII.) will be as -

SECTION IV.

Of the finding of elliptic, parabolic, and hyperbolic orbits, from ttu.
focus given.

LEMMA XV.

If from the two foci S, II, of any ellipsis or hyberbola, we draw to any
third point V the right lines SV, H V, whereof one HV is equal to the
principal axis of the figure, thai is, to the axis in which the foci are
situated, the other, SV, is bisected in T by t/ie perpendicular TR let
fall upon it ; that perpendicular TR will somewhere touch the conic
section : and, vice versa, if it does touch it, HV will be equal to the
principal axis of the figure.
For, let the perpendicular TR cut the right line
HV, produced, if need be, in R ; and join SR. Be
cause TS, TV are equal, therefore the right lines SR,
VR, as well as the angles TRS, TRV, will be also
equal. Whence the point R will be in the conic section, and the perpen
dicular TR will touch the same ; and the contrary. Q.E.D.

126 THE MATHEMATICAL PBINCIP, -ES [BOOK 1

PROPOSITION XVIII. PROBLEM X.

From a focus and the principal axes given, to describe elliptic and hy
perbolic trajectories, which shall pass through given points, and touch
right lines given by position.
Let S be the common focus of the figures ; AB A 33

the length of the principal axis of any trajectory ; r p T~*

P a point through which the trajectory should \ /R

pass ; and TR a right line which it should touch. / \

About the centre P, with the interval AB — SP, £\ S ~~yf

if the orbit is an ellipsis, or AB {- SP, if the y> G ^

orbit is an hyperbola, describe the circle HG. On the tangent TR let fall
the perpendicular ST, and produce the same to V, so that TV may be
equal to ST; and about V as a centre with the interval AB describe the
circle FH. In this manner, whether two points P, p, are given, or two
tangents TR, tr, or a point P and a tangent TR, we are to describe two
circles. Let H be their common intersection, and from the foci S, H, with
the given axis describe the trajectory : I say, the thing is done. For (be
cause PH -f- SP in the ellipsis, and PH — SP in the hyperbola, is equal
to the axis) the described trajectory will pass through the point P, and (by
the preceding Lemma) will touch the right line TR. And by the same
argument it will either pass through the two points P, p, or touch the two
right lines TR, tr. Q.E.F.

PROPOSITION XIX. PROBLEM XI.

About a given focus, to describe a parabolic trajectory, which shall pass
through given points, and touch right lines given by position.
Let S be the focus, P a point, and TR a tangent of
the trajectory to be described. About P as a centre,
with the interval PS, describe the circle FG. From
the focus let fall ST perpendicular on the tangent, and
produce the same to V, so as TV may be equal to ST.
After the same manner another circle fg is to be de
scribed, if another point p is given ; or another point v
is to be found, if another tangent tr is given; then draw
the right line IF, which shall touch the two circles YG,fg, if two points
P, p are given ; or pass through the two points V, v, if two tangents TR,
tr, are given : or touch the circle FG, and pass through the point V, if the
point P and the tangent TR are given. On FI let fall the perpendicular
SI, and bisect the same in K ; and with the axis SK and principal vertex K
describe a parabola : I say the thing is done. For this parabola (because
SK is equal to IK, and SP to FP) will pass through the point P ; and

/KS

SEC. IV.] OF NATURAL PHILOSOPHY. 127

(by Cor. 3, Lem. XIV) because ST is equal to TV. and STR a light an
gle, it will touch the right line TR. Q.E.F.

PROPOSITION XX. PROBLEM XII.

About a given focus to describe any trajectory given in specie which shah
pass through given points, and touch right lines given by position.
CASE 1. About the focus S it is re-
uired to describe a trajectory ABC, pass
ing through two points B, C. Because the
trajectory is given in specie, the ratio of the

principal axis to the distance of the foci GAS H

will be given. In that ratio take KB to BS, and LC to CS. About the
centres B, C, with the intervals BK, CL, describe two circles ; and on the
right line KL, that touches the same in K and L, let fall the perpendicu
lar SG ; which cut in A and a, so that GA may be to AS, and Ga to aS,
as KB to BS ; and with the axis A.«, and vertices A, a, describe a trajectory :
I say the thing is done. For let H be the other focus of the described
figure, and seeing GA is to AS as Ga to aS, then by division we shall
have Ga — GA, or Aa to «S — AS, or SH in the same ratio; and therefore
in the ratio which the principal axis of the figure to be described has to
the distance of its foci ; and therefore the described figure is of the same
species with the figure which was to be described. And since KB to BS,
and LC to CS, are in the same ratio, this figure will pass through tht-
points B, C, as is manifest from the conic sections.

CASE 2. About the focus S it is required to
describe a trajectory which shall somewhere
touch two right lines TR, tr. From the focus
on those tangents let fall the perpendiculars
ST, St, which produce to V, v, so that TV, tv
may be equal to TS, tS. Bisect Vv in O, and j
erect the indefinite perpendicular OH, and cut I.
the right line VS infinitely produced in K and V
k, so that VK be to KS, and VA* to A~S, as the principal axis of the tra
jectory to be described is to the distance of its foci. On the diameter
K/J describe a circle cutting OH in H ; and with the foci S, H, and
principal axis equal to VH, describe a trajectory : I say, the thing is done.
For bisecting Kk in X, and joining HX, HS, HV, Hv, because VK is to
KS as VA- to A*S ; and by composition, as VK -f- V/c to KS + kS ; and
by division, as VA* — VK to kS — KS, that is, as 2VX to 2KX, and
2KX to 2SX, and therefore as VX to HX and HX to SX, the triangles
VXH, HXS will be similar ; therefore VH will be to SH as VX to XH ;
and therefore as VK to KS. Wherefore VH, the principal axis of the
described trajectory, has the same ratio to SH, the distance of the foci, as

12S

THE MATHEMATICAL PRINCIPLES

[BOOK 1.

K S

the principal axis of the trajectory which was to be described has to the
distance of its foci ; and is therefore of the same species. Arid seeing VH,
vH are equal to the principal axis, and VS, vS are perpendicularly bisected
by the right lines TR, tr, it is evident (by Lem. XV) that those right
lines touch the described trajectory. Q,.E.F.

CASE. 3. About the focus S it is required to describe a trajectory, which
shall touch a right line TR in a given Point R. On the right line TR
Jet fall the perpendicular ST, which produce to V, so that TV may be
equal to ST ; join VR, and cut the right line VS indefinitely produced
in K and k, so. that VK may be to SK, and V& to SAr, as the principal
axis of the ellipsis to be described to the distance of its foci ; and on the
diameter KA: describing a circle, cut the H

right line VR produced in H ; then with
the foci S, H, and principal axis equal to R
VH, describe a trajectory : I say, the thing .---'
is done. For VH is to SH as VK to SK, V" "1
and therefore as the principal axis of the trajectory which was to be de
scribed to the distance of its foci (as appears from what we have demon
strated in Case 2) ; and therefore the described trajectory is of the same
species with that which was to be described ; but that the right line TR,
by which the angle VRS is bisected, touches the trajectory in the point R,
is certain from the properties of the conic sections. Q.E.F.

CASE 4. About the focus S it is r

required to describe a trajectory
APB that shall touch a right line
TR, and pass through any given
point P without the tangent, and
shall be similar to the figure apb,
described with the principal axis ab,
and foci s, h. On the tangent TR
let fall the perpendicular ST, which / ..''''',.---"'"
produce to V, so that TV may be
equal to ST ; and making the an
gles hsq, shq, equal to the angles VSP, SVP, about q as a centre, and
with an interval which shall be to ab as SP to VS, describe a circle cut
ting the figure apb in p : join sp, and draw
SH such that it may be to sh as SP is to sp,
and may make the angle PSH equal to the
angle psh, and the angle VSH equal to the
angle pyq. Then with the foci S, H, and B
principal axis AB, equal to the distance VH,
describe a conic section : I say, the thing is
done ; for if sv is drawn so that it shall be to

SEC. IV.] OF NATURAL PHILOSOPHY. 129

sp as sh is to sq, and shall make the angle vsp equal to the angle hsq, and
the angle vsh equal to the angle psq, the triangles svh, spq, will be similar,
and therefore vh will be to pq as sh is to sq ; that is (because of the simi
lar triangles VSP, hsq), as VS is to SP? or as ab to pq. Wherefore
vh and ab are equal. But, because of the similar triangles VSH, vsh, VH
is to SH as vh to sh ; that is, the axis of the conic section now described
is to the distance of its foci as the axis ab to the distance of the foci sh ;
and therefore the figure now described is similar to the figure aph. But,
because the triangle PSH is similar to the triangle psh, this figure passes
through the point P ; and because VH is equal to its axis, and VS is per
pendicularly bisected by the rght line TR, the said figure touches the
right line TR. Q.E.F.

LEMMA XVI.

From three given points to draw to afonrth point that is not given three
right lines whose differences shall be either given, or none at all.
CASE 1. Let the given points be A, B, C, and Z the fourth point which
we are to find ; because of the given difference of the lines AZ, BZ, the
locus of the point Z will be an hyperbola
whose foci are A and B, and whose princi
pal axis is the given difference. Let that
axis be MN. Taking PM to MA as MN
is to AB, erect PR perpendicular to AB,
and let fall ZR perpendicular to PR ; then
from the nature of the hyperbola, ZR will
be to AZ as MN is to AB. And by the
like argument, the locus of the point Z will
be another hyperbola, whose foci are A, C, and whose principal axis is the
difference between AZ and CZ ; and QS a perpendicular on AC may be
drawn, to which (QS) if from any point Z of this hyperbola a perpendicular
ZS is let fall (this ZS), shall be to AZ as the difference between AZ and
CZ is to AC. Wherefore the ratios of ZR and ZS to AZ are given, and
consequently the ratio of ZR to ZS one to the other ; and therefore if the
right lines RP, SQ, meet in T, and TZ and TA are drawn, the figure
TRZS will be given in specie, and the right line TZ, in which the point
Z is somewhere placed, will be given in position. There will be given
also the right line TA, and the angle ATZ ; and because the ratios of AZ
and TZ to ZS are given, their ratio to each other is given also ; and
thence will be given likewise the triangle ATZ, whose vertex is the point
Z. Q.E.I.

CASE 2. If two of the three lines, for example AZ and BZ, are equal,
draw the right line TZ so as to bisect the right line AB ; then find the
triangle ATZ as above. Q.E.I.

130

THE MATHEMATICAL PRINCIPLES

[BOOK I.

CASE 3. If all the three are equal, the point Z will be placed in the
centre of a circle that passes through the points A, B, C. Q.E.I.

This problematic Lemma is likewise solved in Apollonius's Book oi
Tactions restored by Vieta.

PROPOSITION XXL PROBLEM XIII.

About a given focus to describe a trajectory that shall pass through
given points and touch right Hues given by position.
Let the focus S, the point P, and the tangent TR be given, and suppose

that the other focus H is to be found.

On the tangent let fall the perpendicular

ST, which produce to Y, so that TY may

be equal to ST, and YH will be equal

to the principal axis. Join SP, HP, and

SP will be the difference between HP and

the principal axis. After this manner,

if more tangents TR are given, or more

points P. we shall always determine as

many lines YH, or PH, drawn from the said points Y or P, to the focus

H, which either shall be equal to the axes, or differ from the axes by given

lengths SP ; and therefore which shall either be equal among themselves,

or shall have given differences ; from whence (by the preceding Lemma).

that other focus H is given. But having the foci and the length of the

axis (which is either YH, or, if the trajectory be an ellipsis, PH -f SP ;

or PH — SP, if it be an hyperbola), the trajectory is given. Q.E.I.

SCHOLIUM.

When the trajectory is an hyperbola, I do not comprehend its conjugate
hyperbola under the name of tins trajectory. For a body going on with a
continued motion can never pass out of one hyperbola into its conjugate
hyperbola.

The case when three points are given
is more readily solved thus. Let B, C,
I), be the given points. Join BC, CD,
and produce them to E, F, so as EB may
be to EC as SB to SC ; and FC to FD
as SC to SD. On EF drawn and pro
duced let fall the perpendiculars SG,
BH, and in GS produced indefinitely E
take GA to AS, and Ga to aS, as HB
is to BS ; then A will be the vertex, and Aa the principal axis of the tra
jectory ; which, according as GA is greater than, equal to, or less than

SEC. V.] OF NATURAL PHILOSOPHY. 131

AS. will be either an ellipsis, a parabola, or an hyperbola ; the point a in
the first case falling on the same side of the line GP as the point A ; in
the second, going oft* to an infinite distance ; in the third, falling on the
other side of the line GP. For if on GF the perpendiculars CI, DK are
let fall, TC will be to HB as EC to EB ; that is, as SO to SB ; and by
permutation, 1C to SC as HB to SB, or as GA to SA. And, by the like
argument, we may prove that KD is to SD in the same ratio. Where
fore the points B, C, D lie in a conic section described about the focus S,
in such manner that all the right lines drawn from the focus S to the
several points of the section, and the perpendiculars let fall from the same
points on the right line GF, are in that given ratio.

That excellent geometer M. De la Hire has solved this Problem much
after the same way, in his Conies, Prop. XXV., Lib. VIII.

SECTION V.

How the orbits are to be found when neither focus is given.

LEMMA XVII.

If from any point P of a given conic section, to the four produced sides
AB, CD, AC, DB, of any trapezium ABDC inscribed in that section,
as many right lines PQ, PR, PS, PT are drawn in given ang7ei,
each line to each side ; the rectangle PQ, X PR of those on the opposite
sides AB, CD, will be to the rectangle PS X PT of those on tie other
two opposite sides AC, BD, in a given ratio.
CASE 1. Let us suppose, first, that the lines drawn

to one pair of opposite sides are parallel to either of I ^^ p ;T
the other sides ; as PQ and PR to the side AC, and s|
PS and PT to the side AB. And farther, that one
pair of the opposite sides, as AC and BD, are parallel
betwixt themselves; then the right line which bisects^ IQ I3

those parallel sides will be one of the diameters of the 1L

conic section, and will likewise bisect RQ. Let O be the point in which
RQ is bisected, and PO will be an ordinate to that diameter. Produce
PO to K, so that OK may be equal to PO, and OK will be an ordinate
on the other side of that diameter. Since, therefore, the points A, B; P
and K are placed in the conic section, and PK cuts AB in a given angle,
the rectangle PQK (by Prop. XVII., XIX., XXI. and XXI1L, Book III.,
of Apollonius's Conies) will be to the rectangle AQB in a given ratio.
But QK and PR are equal, as being the differences of the equal lines OK,
OP, and OQ, OR ; whence the rectangles PQK and PQ X PR are equal ;
and therefore the rectangle PQ X PR is to the rectangle A^ B, that Is, to
the rectangle PS X PT in a given ratio. Q.E.D

132

THE MATHEMATICAL PRINCIPLES

[BOOK I

CASE 2. Let us next suppose that the oppo
site sides AC and BD of the trapezium are not
parallel. Draw Be/ parallel to AC, and meeting
as well the right line ST in /, as the conic section
in d. Join Cd cutting PQ in r, and draw DM
parallel to PQ, cutting Cd in M, and AB in N.

Then (because of the similar triangles BTt,

DBN), Et or PQ is to Tt as DN to NB. And ^^ Q N

so Rr is to AQ or PS as DM to AN. Wherefore, by multiplying the antece-
dents by the antecedents, and the consequents by the consequents, as the
rectangle PQ X Rr is to the rectangle PS X Tt, so will the rectangle
N i)M be to the rectangle ANB ; and (by Case 1) so is the rectangle
PQ X Pr to the rectangle PS X Pt : and by division, so is the rectangle
PQ X PR to the rectangle PS X PT. Q.E.D.

CASE 3. Let us suppose, lastly, the four lines
?Q, PR, PS, PT, not to be parallel to the sides
AC, AB, but any way inclined to them. In their
place draw Pq, Pr, parallel to AC ; and Ps, Pt
parallel to AB ; and because the angles of the
triangles PQ</, PRr, PSs, PTt are given, the ra-
tios of IQ to Pq, PR to Pr, PS to P*, PT to Pt
will b? also given; and therefore the compound
ed ratios Pk X PR to P? X Pr, and PS X PT to Ps X Pt are
given. But from what we have demonstrated before, the ratio of Pq X Pi
to Ps X Pt is given ; and therefore also the ratio of PQ X PR to PS X
PT. Q.E.D.

LEMMA XVIII.

The s 'niL things supposed, if the rectangle PQ X PR of the lines drawn
to the two opposite sides of the trapezium is to the rectangle PS X PT
of those drawn to the other two sides in a given ratio, the point P,
from whence those lines are drawn, will be placed in a conic section
described about the trapezium.
Conceive a conic section to be described pas
sing through the points A, B, C, D, and any

one of the infinite number of points P, as for

example p ; I say, the point P will be always c1

placed in this section. If you deny the thing,

join AP cutting this conic section somewhere

else, if possible, than in P, as in b. Therefore

if from those points p and b, in the given angles ^ B

to the sides of the trapezium, we draw the right

lines pq, pr, ps, pt, and bk, bn, bf, bd, we shall have, as bk X bn to bf X bd,

SEC. V.] OF NATURAL PHILOSOPHY 133

so (by Lem. XVII) pq X pr to ps X pt ; and so (by supposition) PQ x
PR to PS X PT. And because of the similar trapezia bkAf, PQAS, as
bk to bf, so PQ to PS. Wherefore by dividing the terms of the preceding
proportion by the correspondent terms of this, we shall have bn to bd as
PR to PT. And therefore the equiangular trapezia ~Dnbd, DRPT, are
similar, and consequently their diagonals D6, DP do coincide. Wherefore
b falls in the intersection of the right lines AP, DP, and consequently
coincides with the point P. And therefore the point P, wherever it is
taken, falls to be in the assigned conic section. Q.E.D.

COR. Hence if three right lines PQ, PR, PS, are drawn from a com
mon point P, to as many other right lines given in position, AB, CD, AC,
each to each, in as many angles respectively given, and the rectangle PQ
X PR under any two of the lines drawn be to the square of the third PS
in a given ratio ; the point P, from which the right lines are drawn, will
be placed in a conic section that touches the lines AB; CD in A and C •
and the contrary. For the position of the three right lines AB, CD, AC
remaining the same, let the line BD approach to and coincide with the
line AC ; then let the line PT come likewise to coincide with the line PS ;
and the rectangle PS X PT will become PS2, and the right lines AB, CD,
which before did cut the curve in the points A and B, C and D, can no
(onger cut, but only touch, the curve in those coinciding points.

SCHOLIUM.

In this Lemma, the name of conic section is to be understood in a large
sense, comprehending as well the rectilinear section through the vertex of
the cone, as the circular one parallel to the base. For if the point p hap
pens to be in a right line, by which the points A and D, or C and B are
joined, the conic section will be changed into two right lines, one of which
is that right line upon which the point p falls,
and the other is a right line that joins the other
two of *he four points. If the two opposite an
gles of the trapezium taken together are equal c
to two right angles, and if the four lines PQ,
PR, PS, PT, are drawn to the sides thereof at
right angles, or any other equal angles, and the
rectangle PQ X PR under two of the lines
drawn PQ and PR, is equal to the rectangle
PS X PT under the other two PS and PT, the conic section will become
a circle. And the same thing will happen if the four lines are drawn in
any angles, and the rectangle PQ X PR, under one pair of the lines drawn,
is to the rectangle PS X PT under the other pair as the rectangle under
the sines of the angles S, T, in which the two last lines PS, PT are drawn
to the rectangle under the sines of the angles Q, R, in which the first tw«

134 THE MATHEMATICAL PRINCIPLES [BOOK 1.

PQ, PR are drawn. In all other cases the locus of the point P will be
one of the three figures which pass commonly by the name of the conic
sections. But in room of the trapezium A BCD, we may substitute a
quadrilateral figure whose two opposite sides cross one another like diago
nals. And one or two of the four points A, B, C, D may be supposed to
be removed to an infinite distance, by which means the sides of the figure
which converge to those points, will become parallel ; and in this case the
conic section will pass through the other points, and will go the same way
as the parallels in, infinitum.

LEMMA XIX.

To find a point P from which if four right lines PQ, PR, PS, PT an
drawn to as many other right lines AB, CD, AC, BD, given by posi
tion, each to each, at given angles, the rectangle PQ X PR, under any
two of the lines' drawn, shall be to the rectangle PS X PT, under the
other tivo. in a given ratio.
Suppose the lines AB, CD, to which the two
right lines PQ, PR, containing one of the rect
angles, are drawn to meet two other lines, given
by position, in the points A, B, C, D. From one
of those, as A, draw any right line AH, in which
you would find the point P. Let this cut the
opposite lines BD, CD, in H and I ; and, because
all the angles of the figure are given, the ratio of
PQ to PA, and PA to PS, and therefore of PQ

to PS, will be also given. Subducting this ratio from the given ratio oi
PQ X PR to PS X PT, the ratio of PR to PT will be given ; and ad
ding the given ratios of PI to PR, and PT to PH, the ratio of PI to PH.
and therefore the point P will be given. Q.E.I.

COR. 1. Hence also a tangent may be drawn to any point D of the
locus of all the points P. For the chord PD, where the points P and D
meet, that is, where AH is drawn through the point D, becomes a tangent.
In which case the ultimate ratio of the evanescent lines IP and PH will
be found as above. Therefore draw CF parallel to AD, meeting BD in
F, and cut it in E in the same ultimate ratio, then DE will be the tan
gent ; because CF and the evanescent IH are parallel, and similarly cut in
E and P.

COR. 2. Hence also the locus of all the points P may be determined.
Through any of the points A, B, C, D, as A, draw AE touching the locus,
and through any other point B parallel to the tangent, draw BF meeting
the locus in F ; and find the point F by this Lemma. Bisect BF in G,
and, drawing the indefinite line AG, this will be the position of the dia
meter to which BG and FG are ordinates. Let this AG meet the locus

SEC. V.J

OF NATURAL PHILOSOPHY.

in H, and AH will be its diameter or latus trans-
versum. to which the latus rectum will be as BG2
to AG X GH. If AG nowhere meets the locus,
the line AH being infinite, the locus will be a par
abola ; and its latus rectum corresponding to the

diameter AG will be -.-7^
AC*

But if it does meet it

anywhere, the locus will be an hyperbola, when

the points A and H are placed on the same side the point G ; and an
ellipsis, if the point G falls between the points A and H ; unless, perhaps,
the angle AGB is a right angle, and at the same time BG2 equal to the
rectangle AGH, in which case the locus will be a circle.

And so we have given in this Corollary a solution of that famous Prob
lem of the ancients concerning four lines, begun by Euclid, and carried on
by Apollonius ; and this not an analytical calculus, but a geometrical com
position, such as the ancients required.

LEMMA XX.

If the two opposite angular points A and P of any parallelogram ASPQ
touch any conic section in the points A and P ; and the sides AQ, AS
of one of those angles, indefinitely produced, meet the same conic section
in B and C ; and from the points of concourse, B and C to any fifth
point D of the conic section, two right lines BD, CD are drawn meet-
ing tlie two other sides PS, PQ of the parallelogram, indefinitely pro
duced in T and R ; the parts PR and PT, cut off from the sides, will
always be one to the other in a given ratio. And vice versa, if those
parts cut off' are one to the other in a given ratio, the locus of the point
D will be a conic section passing through the four points A, B, C, F
CASE 1. Join BP, CP, and from the point

D draw the two right lines DG, DE, of which

the first DG shall be parallel to AB, and

meet PB, PQ, CA in H, I, G ; and the other

DE shall be parallel to AC, and meet PC,

PS, AB, in F, K, E ; and (by Lem. XVII)

the rectangle DE X DF will be to the rect

angle DG X DH in a given ratio. But

PQ is to DE (or IQ) as PB to HB, and con

sequently as PT to DH ; and by permutation PQ, is to PT as DE to

DH. Likewise PR is to DF as RC to DC, and therefore as (IG or) PS

to DG ; and by permutation PR is to PS as DF to DG ; and, by com

pounding those ratios, the rectangle PQ X PR will be to the rectangle

PS X PT as the rectangle DE X DF is to the rectangle DG X DH.

and consequently in "a given ratio. But PQ and PS are given, and there

fore the ratio of PR to PT is given. Q.E.D.

136

THE MATHEMATICAL PRINCIPLES

CASE 2. But if PR and PT are supposed to be in a given ratio one to
the other, then by going back again, by a like reasoning, it will follow
that the rectangle DE X DF is to the rectangle DG X DH in a given
rati) ; and so the point D (by Lem. XVIII) will lie in a conic section pass
ing through the points A., B, C, P, as its locus. Q.E.I).

COR. 1. Hence if we draw BC cutting PQ in r and in PT take Pt to
Pr in the same ratio which PT has to PR ; then Et will touch the conic
section in the point B. For suppose the point D to coalesce with the point
B, so that the chord BD vanishing, BT shall become a tangent, and CD
and BT will coincide with CB and Bt.

COR. 2. And, vice versa, if Bt is a tangent, and the lines BD, CD meet
in any point D of a conic section, PR will be to PT as Pr to Pt. And,
on the contrary, if PR is to PT as Pr to Pt, then BD and CD will meet
in some point D of a conic section.

COR. 3. One conic section cannot cut another conic section in more than
four points. For, if it is possible, let two conic sections pass through the
h've points A, B, C, P, O ; and let the right line BD cut them in the
points D, d, and the right line Cd cut the right line PQ, in q. Therefore
PR is to PT as Pq to PT : whence PR and Pq are equal one to the other,
against the supposition.

LEMMA XXI.

If two moveable and indefinite right lines BM, CM drawn through given
points B, C, as poles, do by their point of concourse M describe a third
right line MN given by position ; and other two indefinite right lines
BD,CD are drawn, making with the former two at those given points
B, C, given angles, MBD, MCD : I say, that those two right lines BD,
CD will by their point of concourse D describe a conic section passing
through the points B, C. And, vice versa, if the right lints BD, CD
do by their point of concourse D describe a conic section passing
through the given points B, C, A, and the angle DBM is always
equal to the giren angle ABC, as well as the angle DCM always
equal to the given angle ACB, the point M will lie in a right line
given by position, as its locus.
For in the right line MN let a point

N be given, and when the moveable point

M falls on the immoveable point N. let

the moveable point D fall on an immo
vable point P. Join ON, BN, CP, BP,

and from the point P draw the right lines

PT, PR meeting BD, CD in T and R, C

and making the angle BPT c jual to the

given angle BNM, and the angle CPR

SEC. V.J

OF NATURAL PHILOSOPHY.

137

equal to the given angle CNM. Wherefore since (by supposition) the an
gles MBD, NBP are equal, as also the angles MOD, NCP, take away the
angles NBD and NOD that are common, and there will remain the angles
NBM and PBT, NCM and PCR equal; and therefore the triangles NBM,
PBT are similar, as also the triangles NCM, PCR. Wherefore PT is to
NM as PB to NB ; and PR to NM as PC to NC. But the points, B, C,
N, P are immovable: wheiefore PT and PR have a given ratio to NM,
and consequently a given ratio between themselves; and therefore, (by
Lemma XX) the point D wherein the moveable right lines BT and CR
perpetually concur, will be placed in a conic section passing through the
points B. C, P. Q.E.D.

And, vice versa, if the moveable point
D lies in a conic section passing through
the given points B, C, A ; and the angle
DBM is always equal to the given an
gle ABC, and the angle DCM always
equal to the given angle ACB, and when
the point D falls successively on any
two immovable points p, P, of the conic
section, the moveable point M falls suc
cessively on two immovable points /?, N.
Through these points ??, N, draw the right line nN : this line nN will be
the perpetual locus of that moveable point M. For, if possible, let the
point M be placed in any curve line. Therefore the point D will be placed
in a conic section passing through the five points B, C, A, p, P, when the
point M is perpetually placed in a curve line. But from what was de
monstrated before, the point D will be also placed in a conic section pass
ing through the same five points B, C, A, p, P, when the point M is per
petually placed in a right line. Wherefore the two conic sections will both
pass through the same five points, against Corol. 3, Lem. XX. It is
therefore absurd to suppose that the point M is placed in a curve line.
QE.D.

PROPOSITION XXII. PROBLEM XIV.

To describe a trajectory that shall pass through Jive given points.
Let the five given points be A, B, C, P, D. c
From any one of them, as A, to any other sv
two as B, C, which may be called the poles,
draw the right lines AB, AC, and parallel to
those the lines TPS, PRO, through the fourth
point P. Then from the two poles B, C,
draw through the fifth point D two indefinite
lines BDT, CRD, meeting with the last drawn lines TPS, PRQ (the

138

THE MATHEMATICAL PRINCIPLES

IBOOK L

former with the former, and the latter with the latter) in T and R. Then
drawing the right line tr parallel to TR, cutting off from the right lines
PT, PR, any segments Pt, Pr, proportional to PT, PR ; and if through
their extremities, t, r, and the poles B, C, the right lines lit, Cr are drawn,
meeting in d, that point d will be placed in the trajectory required. For
(by Lena. XX) that point d is placed in a conic section passing through
the four points A, B, C, P ; and the lines R/', TV vanishing, the point d
comes to coincide with the point D. Wherefore the conic section passes
through the five points A, B, C, P, D. Q.E.D.

The same otherwise.

Of the given points join any three, as A, B,
C ; and about two of them 15, C, as poles,
making the angles ABC, ACB of a given
magnitude to revolve, apply the legs BA,
CA, first to the point D, then to the point P,
and mark the points M, N, in which the other
legs BL, CL intersect each other in both cases. C'
Draw the indefinite right line MN, and let
those moveable angles revolve about their
poles B, C, in such manner that the intersection, which is now supposed to
be ???, of the legs BL, CL; or BM7 CM, may always fall in that indefinite
right line MN ; and the intersection, which is now supposed to be d, of the
legs BA ^A, or BD; CD, will describe the trajectory required, PADc/B.
For (by Lem. XXI) the point d will be placed in a conic section passing-
through the points B, C ; and when the point m comes to coincide with
the points L, M, N, the point d will (by construction) come to coin
cide with the points A, D, P. Wherefore a conic section will be described
that shall pass through the five points A, B. C, P, D. Q,.E.F.

COR. 1. Hence a right line may be readily drawn which shall be a tan
gent to the trajectory in any given point B. Let the point d come to co
incide with the point B, arid the right line Bt/ Avill become the tangent
required.

COR. 2. Hence also may be found the centres, diameters, and latera recta
of the trajectories, as in Cor. 2, Lem. XIX.

SCHOLIUM.

The former of these constructions will be- c
come something more simple by joining ,
and in that line, produced, if need be, aking
Bp to BP as PR is to PT ; and t rough p
draw the indefinite right inc j0e parallel to S
PT, and in that line pe taking always pe
equal to Pi , and draw the right lines Be, Cr

SEC. Y.J

OF NATURAL PHILOSOPHY.

139

to meet in d. For since Pr to Pt, PR to PT, pB to PB, pe to Pt, are all in

the same ratio, pe and Pr will be always equal. After this manner the
points of the trajectory are most readily found, unless you would rather
describe the curve mechanically, as in the second construction.

PROPOSITION XXIII. PROBLEM XV.

To describe a trajectory that shall pass through four given points, and

touch a right line given by position.

CASE 1. Suppose that HB is the
given tangent, B the point of contact,
and C, 1., P, the three other given
points. Jo n BC. and draw IS paral
lel to BH, and PQ parallel to BC ;
complete the parallelogram BSPQ.
Draw BD cutting SP in T, and CD
cutting PQ, in R. Lastly, draw any
line tr parallel to TR, cutting off
from PQ, PS, the segments Pr, Pt proportional to PR, PT respectively ;
and draw Cr, Bt their point of concourse d will (by Lem. XX) always fall
on the trajectory to be described.

The same otherwise.

1 et tl e angle CBH of a given magnitude re
volve about the pole B; as also the rectilinear ra-
d:us 1C, both ways produced, about the pole C.
Mark the points M, N, on which the leg BC of
the angle cuts that radius when BH; the other
leg thereof, meets the same radius in the points
P and D. Then drawing the indefinite line MN,
let that radius CP or CD and the leg BC of the
angle perpetually meet in this Ikie; and the
point of concourse of the other leg BH with the
radius will delineate the trajectory required.

For if in the constructions of the preceding Problem the point A comes
to a coincidence with the point B, the lines CA and CB will coincide, and
the line AB, in its last situation, will become the tangent BH ; and there
fore the constructions there set down will become the same with the con
structions here described. Wherefore the concourse of the leg BH with
the radius will describe a conic section passing through the points C, D,
P, and touching the line BH in the point B. Q.E.F.

CASE 2. Suppose the four points B, C, D, P, given, being situated with-
ont the tangent HI. Join each two by the lines BD, CP meeting in G,
and cutting the tangent in H and I. Cut the tangent in A in such mannr:

140

THE MATHEMATICAL PRINCIPLES

[BOOK I

X

IT

that HA may be to IA as the rectangle un
der a mean proportional between CG and
GP, and a mean proportional between BH
and HD is to a rectangle under a mean pro
portional between GD and GB, and a mean
proportional betweeen PI and 1C, and A will
be the point of contact. For if HX, a par
allel to the right line PI, cuts the trajectory
in any points X and Y, the point A (by the
properties of the conic sections) will come to be so placed, that HA2 will
become to AP in a ratio that is compounded out of the ratio of the rec
tangle XHY to the rectangle BHD, or of the rectangle CGP to the rec
tangle DGB; and the ratio of the rectangle BHD to the rectangle PIC.
But after the point of contac.t A is found, the trajectory will be described as
in the first Case. Q.E.F. But the point A may be taken either between
or without the points H and I, upon which account a twofold trajectory
may be described.

PROPOSITION XXIV. PROBLEM XVI.

To describe a trajectory that shall pass through three given points, and

touch two right lines given by position.

Suppose HI, KL to be the given tangents
and B, C, D, the given points. Through any
two of those points, as B, D, draw the indefi
nite right line BD meeting the tangents in
the points H, K. Then likewise through
any other two of these points, as C, D, draw
the indefinite right line CD meeting the tan
gents in the points I, L. Cut the lines drawn
in R and S, so that HR may be to KR as
the mean proportional between BH and HD is to the mean proportional
between BK and KD ; and IS to LS as the mean pioportional between
CI and ID is to the mean proportional between CL and LD. But you
may cut, at pleasure, either within or between the points K and H, I and
L, or without them ; then draw RS cutting the tangents in A and P, and
A and P will be the points of contact. For if A and P are supposed to
be the points of contact, situated anywhere else in the tangents, and through
any of the points H, I, K, L, as I, situated in either tangent HI, a right
line IY is drawn parallel to the other tangent KL, and meeting the curve
in X and Y, and in that right line there be taken IZ equal to a mean pro
portional between IX and IY, the rectangle XIY or IZ2, will (by the pro
perties of the conic sections) be to LP2 as the rectangle CID is to the rect
angle CLD, that is (by the construction), as SI is to SL2; and therefore

SEC. V.] OF NATUKAL PHILOSOPHY. 141

IZ is to LP as SI to SL. Wherefore the points S, P, Z. are in one right
line. Moreover, since the tangents meet in G, the rectangle XI Y or IZ2
will (by the properties of the conic sections) be to IA2 as GP2 is to GA2,
and consequently IZ will be to I A as GP to GA. Wherefore the points
P, Z, A, lie in one right line, and therefore the points S, P, and A are in
one right line. And the same argument will prove that the points R, P,
and A are in one right line. Wherefore the points of contact A and P lie
in the right line RS. But after these points are found, the trajectory may
be described, as in the first Case of the preceding Problem. Q,.E.F.

In this Proposition, and Case 2 of the foregoing, the constructions are
the same, whether the right line XY cut the trajectory in X and Y, or
not ; neither do they depend upon that section. But the constructions
being demonstrated where that right line does cut the trajectory, the con
structions where it does not are also known ; and therefore, for brevity's
sake, I omit any farther demonstration of them.

LEMMA XXII.

To transform figures into other figures of the same kind.

Suppose that any figure HGI is to be
transformed. Draw, at pleasure, two par
allel lines AO, BL, cutting any third line
AB, given by position, in A and B, and from
any point G of the figure, draw out any
right line GD, parallel to OA, till it meet
the right line AB. Then from any given
point 0 in the line OA, draw to the point
D the right line OD, meeting BL in d ; and
from the point of concourse raise the right
line dg containing any given angle with the right line BL, and having
such ratio to Qd as DG has to OD ; and g will be the point in the new
figure hgi, corresponding to the point G. And in like manner the several
points of the first figure will give as many correspondent points of the new
figure. If we therefore conceive the point G to be carried along by a con
tinual motion through all the points of the first figure, the point g will
be likewise carried along by a continual motion through all the points of
the new figure, and describe the same. For distinction's sake, let us call
DG the first ordinate, dg the new ordinate, AD the first abscissa, ad the
new abscissa ; O the pole. OD the abscinding radius, OA the first ordinate
radius, and Oa (by which the parallelogram OABa is completed) the new
ordinate radius.

I say, then, that if the point G is placed in a right line given by posi
tion, the point g will be also placed in a right line given by position. If
the point G is placed in a conic section, the point g will be likewise placed

J42 THE MATHEMATICAL PRINCIPLES [BOOK 1.

in a conic section. And here I understand the circle as one of the conic
sections. But farther, if the point G is placed in a line of the third ana
lytical order, the point g will also be placed in a line of the third order,
and so on in curve lines of higher orders. The two lines in which the
points G, g, are placed, will be always of the same analytical order. For
as ad is to OA, so are Od to OD, dg to DG, and AB to AD ; and there-

OA X AB OA X dg

fore AD is equal to , , and DG equal to 7— — . Now if the

ad ad

point G is placed in a right line, and therefore, in any equation by which
the relation between the abscissa AD and the ordinate GD is expressed,
those indetermined lines AD and DG rise no higher than to one dimen-

v v xu- ,. OA X AB . OA X dg

sion, by writing this equation— . m place of AD, and — — -. — -

in place of DG, a new equation will be produced, in which the new ab
scissa ad and new ordinate dg rise only to one dimension ; and which
therefore must denote a right line. But if AD and DG (or either of
them) had risen to two dimensions in the first equation, ad and dg would
likewise have risen to tAvo dimensions in the second equation. And so on
in three or more dimensions. The indetermined lines, ad} dg in the
second equation, and AD, DG, in the first, will always rise to the same
number of dimensions ; and therefore the lines in which the points G, g,
are placed are of the same analytical order.

I say farther, that if any right line touches the curve line in the first
figure, the same right line transferred the same way with the curve into
the new figure will touch that curve line in the new figure, and vice versa.
For if any two points of the curve in the first figure are supposed to ap
proach one the other till they come to coincide, the same points transferred
will approach one the other till they come to coincide in the new figure ;
and therefore the right lines with which those points are joined will be
come together tangents of the curves in both figures. I might have given
demonstrations of these assertions in a more geometrical form ; but I study
to be brief.

Wherefore if one rectilinear figure is to be transformed into another, we
need only transfer the intersections of the right lines of which the first
figure consists, and through the transferred intersections to draw right lines
in the new figure. But if a curvilinear figure is to be transformed, we
must transfer the points, the tangents, and other right lines, by means of
which the curve line is denned. This Lemma is of use in the solution of
the more difficult Problems ; for thereby we maj' transform the proposed
figures, if they are intricate, into others that are more simple. Thus any
right lines converging to a point are transformed into parallels, by taking
for the first ordinate radius any right line that passes through the point
of concourse of the converging lines, and that because their point of con-

SEC. V.] OF NATURAL PHILOSOPHY. 143

course is by this means made to go off in infinitum ; and parallel lines
are such as tend to a point infinitely remote. And after the problem is
solved in the new figure, if by the inverse operations we transform the
new into the first figure, we shall have the solution required.

This Lemma is also of use in the solution of solid problems. For as
often as two conic sections occur, by the intersection of which a problem
may be solved, any one of them may be transformed, if it is an hyperbola
or a parabola, into an ellipsis, and then this ellipsis may be easily changed
into a circle. So also a right line and a conic section, in the construc
tion of plane problems, may be transformed into a right line and a circle

PROPOSITION XXV. PROBLEM XVII.

To describe a trajectory that shall pass through two given points, and

touch three right lines given by position.

Through the concourse of any two of the tangents one with the other,
and the concourse of the third tangent with the right line which passes
through the two given points, draw an indefinite right line ; and, taking
this line for the first ordinate radius, transform the figure by the preceding
Lemma into a new figure. In this figure those two tangents will become
parallel to each other, and the third tangent will be parallel to the right
line that passes through the two given points. Suppose hi, kl to be those
two parallel tangents, ik the third tangent, and hi a right line parallel
thereto, passing through those points a, b,
through which the conic section ought to pass
in this new figure; and completing the paral-
lelogra n fiikl, let the right lines hi, ik, kl be
BO cut in c, d, e, that he may be to the square
root of the rectangle ahb, ic, to id, and ke to
kd. as the sum of the right lines hi and kl is
to the sum of the three lines, the first whereof '
is the right line ik, and the other two are the

square roots of the rectangles ahb and alb ; and c, d, e, will be the points
of contact. For by the properties of the conic sections, he2 to the rectan
gle ahb, and ic2 to id2, and ke2 to kd2, and el2 to the rectangle alb, are all
in the same ratio ; and therefore he to the square root of ahb, ic to id, ke
to kdj and el to the square root of alb, are in the subduplicate of that
ratio ; and by composition, in the given ratio of the sum of all the ante
cedents hi + kly to the sum of all the consequents ^/ahb -\- ik : *Jalb,
Wherefore from that given ratio we have the points of contact c, d, e, in
the new figure. By the inverted operations of the last Lemma, let those
points be transferred into the first figure, and the trajectory will be there
described by Prob. XIV. Q.E.F. But according as the points a, b, fall
between the points //, /, or without taem, the points c, d, e, must be taken

144 THE MATHEMATICAL PRINCIPLES BOOK I.J

Cither between the points, h, i, k, /, or without them. If one of the points
a, b, falls between the points h, i, and the other xvithout the points h, I,
the Problem is impossible.

PROPOSITION XXVI. PROBLEM XVIII.

To describe a trajectory that shall pass through a given point, and touch

four right lines given by position.

From the common intersections, of any two
of the tangents to the common intersection of
the other two, draw an indefinite right line ; and
taking this line for the first ordinate radius; / 'xs o
transform the figure (by Lem. XXII) into a new
figure, and the two pairs of tangents, each of
which before concurred in the first ordinate ra-

dius, will now become parallel. Let hi and kl, Al l\

ik and hi, be those pairs of parallels completing the parallelogram hikl.
And let p be the point in this new figure corresponding to the given point
in the first figure. Through O the centre of the figure draw pq.: and O?
being equal to Op, q will be the other point through which the conic sec
tion must pass in this new figure. Let this point be transferred, by the
inverse operation of Lem. XXII into the first figure, and there we shall
have the two points through which the trajectory is to be described. But
through those points that trajectory may be described by Prop. XVII.

LEMMA XXIII.

If two right lines, as AC, BD given by position, and terminating in

given points A, B, are in a given ratio one to the other, and the right

line CD, by which the, indetermined points C, D are joined is cut in

K in a given ratio ; I say, that the point K will be placed in a right

line given by position.

For let the right lines AC, BD meet in

E, and in BE take BG to AE as BD is to

AC, and let FD be always equal to the given

line EG ; and, by construction, EC will be

to GD, that is, to EF, as AC to BD, and

therefore in a given ratio ; and therefore the %•-'''' ,.---'' I \

triangle EFC will be given in kind. Let E K cT^"^
CF be cut in L so as CL may be to CF in the ratio of CK to CD ; and
because that is a given ratio, the triangle EFL will be given in kind, and
therefore the point L will be placed in the right line EL given by position.
Join LK, and the triangles CLK, CFD will be similar ; and because FD
is a given line, and LK is to FD in a given ratio, LK will be also given

SEC. V.] OF NATURAL PHILOSOPHY. 145

To this let EH be taken equal, and ELKH will be always a parallelogram.
And therefore the point K is always placed in the side HK (given by po
tiition) of that parallelogram. Q.E.D.

COR. Because the figure EFLC is given in kind, the three right lines
EF, EL, and EC, that is, GD, HK, and EC, will have given ratios to
each other.

LEMMA XXIV.

If three right lines, two whereof are parallel, and given by position, touch
any conic section ; I say, that the semi-diameter of the section wkiJt
is parallel to those two is a mean proportional between the segments
of those two that are intercepted between the points of contact and the.
third tangent.

Let AF, GB be the two parallels touch
ing the conic section ADB in A and B ;
EF the third right line touching the conic
section in I, and meeting the two former
tangents in F and G, and let CD be the
semi-diameter of the figure parallel to
those tangents ; I say. that AF, CD, BG
are continually proportional.

For if the conjugate diameters AB, DM G Q

meet the tangent FG in E and H, and cut one the other in C; and the
parallelogram IKCL be completed ; from the nature of the conic sections,
EC will be to CA as CA to CL ; and so by division, EC — CA to CA -—
CL, orEAto AL; and by composition, EA to EA + AL or EL, as EC to
EC + CA or EB ; and therefore (because of the similitude of the triangles
EAF, ELI, ECH, EBG) AF is to LI as CH to BG. Likewise, from tli?
nature of the conic sections, LI (or CK) is to CD as CD to CH ; and
therefore (ex aquo pertnrhatfy AF is to CD as CD to BG. Q.E.D.

COR. 1. Hence if two tangents FG, PQ, meet two parallel tangents AF,
BG in F and G, P and Q,, and cut one the other in O; AF (ex cequo per-
tnrbot, ) will be to BQ as AP to BG, and by division, as FP to GQ, and
therefore as FO to OG.

COR. 2. Whence also the two right lines PG, FQ, drawn through the
points P and G, F and Q, will meet in the right line ACB passing through
the centre of the figure and the points of contact A, B.

LEMMA XXV.

If four sides of a parallelogram indefinitely produced touch any conic
section, and are cut by a fifth tangent ; I say, that, taking those seg
ments of any two conterminous sides that terminate in opposite angles

10

146

THE MATHEMATICAL PRINCIPLES

[BooK 1.

of the parallelogram, either segment is to the side from which it is

cut off as that part of the other conterminous side which is intercepted

between the point of contact and the third side is to Uie other segment,

Let the four sides ML, IK, KL, MI,
of the parallelogram ML JK touch the
conic section in A, B, C, I) ; and let the
fifth tangent FQ cut those sides in F,
Q, H, and E : and taking the segments
ME, KQ of the sides Ml, KJ, or the
segments KH, MF of the sides KL,
ML, 1 s/.y, that ME is to MI as BK to
KQ; and KH to KL as AM to MF.
For, by Cor. 1 of the preceding Lemma, ME is to El as (AM or) BK to
BQ ; and, by composition, ME is to MI as BK to KQ. Q.E.D. Also
KH is to HL as (BK or) AM to AF ; and by division, KH to KL as AM
to MF. Q.E.D.

COR. 1. Hence if a parallelogram IKLM described about a given conic
section is given, the rectangle KQ X ME, as also the rectangle KH X ME
equal thereto, will be given. For, by reason of the similar triangles KQH
MFE, those rectangles are equal.

COR. 2. And if a sixth tangent eq is drawn meeting the tangents Kl.
MI in q and e, the rectangle KQ X ME will be equal to the rectangle
K</ X Me, and KQ will be to Me as Kq to ME, and by division ns
Q? to Ee.

COR. 3. Hence, also, if E<?, eQ, are joined and bisected, and a right line
is drawn through the points of bisection, this right line will pass through
the centre of the conic section. For since Q</ is to Ee as KQ to Me, the
same right line will pass through the middle of all the lines Eq, eQ, MK
(by Lem. XXIII), and the middle point of the right line MK is the
centre of the section.

PROPOSITION XXVII. PROBLEM XIX.

To describe a trajectory that may touch jive right lines given by position.

Supposing ABG; BCF,
GCD, FDE, EA to be the

tangents given by position.
Bisect in M and N, AF, BE,
the diagonals of the quadri
lateral
tained

figure
under

ABFE con-
any four of
them ; and (by Cor. 3, Lem.
XXV) the right line MN
draAvn through the points (,f

SEC. V.] OF NATURAL PHILOSOPHY. 147

bisection will pass through the centre of the trajectory. Again, bisect in
P and Q, the diagonals (if I may so call them) Bl), GF of the quadrila
teral figure EC OF contained under any other four tangents, and the right
line PQ, drawn through the points of bisection will pass through the cen
tre of the trajectory ; and therefore the centre will be given in the con
course of the bisecting lines. Suppose it to be O. Parallel to any tan
gent BG draw KL at such distance that the centre O may be placed in the
middle between the parallels; this KL will touch the trajectory to be de
scribed. Let this cut any other two tangents GCD, FJ)E, in L and K.
Through the points G and K, F and L, where the tangents not parallel,
CL, FK meet the parallel tangents CF, KL, draw GK, FL meeting in
K ; and the right line OR drawn and produced, will cut the parallel tan
gents GF, KL, in the points of contact. This appears from Gor. 2, Lem.
XXIV. And by the same method the other points of contact may be
found, and then the trajectory may be described by Prob. XIV. Q.E.F.

SCPIOLTUM.

Under the preceding Propositions are comprehended those Problems
wherein either the centres or asymptotes of the trajectories are given. For
when points and tangents and the centre are given, as many other points
and as many other tangents are given at an equal distance on the other
side of the centre. And an asymptote is to be considered as a tangent, ami
its infinitely remote extremity (if we may say so) is a point of contact.
Conceive the point of contact of any tangent removed in infinitum, and
the tangent will degenerate into an asymptote, and the constructions of
the preceding Problems will be changed into the constructions of those
Problems wherein the asymptote is given.

After the trajectory is described, we may
find its axes and foci in this manmr. In the
construction and figure of Lem. XXI, let those ,

legs BP, CP, of the moveable angles PEN, ^
PCN, by the concourse of which the trajec- \
tory was described, be made parallel one to
the other : and retaining that position, let
them revolve about their poles I', C, in that
figure. In the mean while let the other legs
GN, BN, of those angles, by their concourse
K or k, describe the circle BKGC. Let O be the centre of this circle;
and from this centre upon the ruler MN, wherein those legs CN, BN did
concur while the trajectory was described, let fall the perpendicular OH
meeting the circle in K and L. And when those other legs CK, BK meet
in the point K that is nearest to the ruler, the first legs CP, BP will be
parallel to the greater axis, and perpendicular on the lesser ; and the con-

148

THE MATHEMATICAL PRINCIPLES

[Book I

trary will hajpen if those legs meet in the remotest point L. 'Whence ii
the centre of the trajectory is given, the axes will be given ; and those be-
ing given, the foci will be readily found.

But the squares of the axes are one to the
other as KH to LH, and thence it is easy to
describe a trajectory given in kind through
f mr given points. For if two of the given
points are made the poles C, 13, the third will
give the moveable angles PCK, PBK ; but
those being given, the circle BGKC may be
described. Then, because the trajectory is
given in kind, the ratio of OH to OK, and
and therefore OH itself, will be given. About '
the centre O, with the interval OH, describe another circle, and the right
line that touches this circle, and passes through the concourse of the legs
CK, BK, when the first legs CP; BP meet in the fourth given point, will
be the ruler MN, by means of which the trajectory may be described
Whence also on the other hand a trapezium given in kind (excepting a
few cases that are impossible) may be inscribed in a given conic section.

There are also other Lemmas, by the help of which trajectories given in
kind may be described through given points, and touching given lines.
Of such a sort is this, that if a right line is drawn through any point
given by position, that may cut a given conic section in two points, and
the distance of the intersections is bisected, the point of bisection will
to'ich ano her conic section of the same kind with the former, arid havin^
its axes parallel to the axes of the former. But I hasten to things of
greater use.

LEMMA XXVI.

To place 1ht' lit rev angles of a triangle, given both in kind and magni
tude, in, respect of as many rigid lines given by position, -provided th°\]
are not all parallel among themselves, in such manner tfia' tjic spiral
angles may touch the several lines.

Three indefinite right lines AB, AC, BC, are
given by position, and it is required so to place
the triangle DEF that its angle 1) may touch
the line AB, its angle E the line AC, and
its angle F the line BC. Upon DE, DF, and
EF, describe three segments of circles DRE,
DGF. EMF, capable of angles equal to the
Rubles BAG, ABC, ACB respectively. But those segments are to be de
scribed t« wards such sides of the lines DE, DF; EF; that the letters

3 EC. V.I

OF NATURAL PHILOSOPHY.

1411

DRED may turn round about in the same order with the letters I1ACB :
the letters DGFD in the same order with the letters ABCA ; and the
letters EMFE in the same order with the letters ACBA ; then, completing
th 'se segmerts into entire circles let the two former circles cut one the
other in G, and suppose P and Q to be their centres. Then joining GP,
PQ, take Ga to AB as GP is to PQ ; and about the centre G, with the
interval Ga, describe a circle that may cut the first circle DGE in a.
Join aD cutting the second circle DFG in b, as well as aE cutting the
third circle EMF in c. Complete the figure ABCdef similar and equal
to the figure a&cDEF : I say, the thing is done.

For drawing Fc meeting «D in n,
and joining aG; bG, QG, QD. PD, by
construction the angle EaD is equal to
the angle CAB, and the angle acF equal
to the angle ACB; and therefore the
triangle aiic equiangular to the triangle
ABC. Wherefore the angle anc or FnD
is equal to the angle ABC, and conse-
< uently to the angle F/>D ; and there
fore the point n falls on the point b,
Moreover the angle GPQ, which is half
the angle GPD at the centre, is equal
to the angle GaD at the circumference \
and the angle GQP, which is half the angle GQD at the centre, is equal
to the complement to two right angles of the angle GbD at the circum
ference, and therefore equal to the angle Gba. Upon which account the
triangles GPQ, Gab, are similar, and Ga is to ab as GP to PQ. ; that is
(by construction), as Ga to AB. Wherefore ab and AB are equal; and
consequently the triangles abc, ABC, which we have now proved to be
similar, are also equal. And therefore since the angles I), E, F, of the
triangle DEF do respectively touch the sides ab, ar, be of the triangle
afjc/ the figure AECdef may be completed similar and equal to the figure
afrcDEFj and by completing it the Problem will be solved. Q.E.F.

COR. Hence a right line may be drawn whose parts given in length may
be intercepted between three right lines given by position. Suppose the
triangle DEF, by the access of its point D to the side EF, arid by having
the sides DE, DF placed i>t directum to be changed into a right line
whose given part DE is to be interposed between the right lines AB; AC
given by position; and its given part DF is to be interposed between the
right lines AB; BC, given by position; then, by applying the preceding
construction to this case, the Problem will be solved.

THE MATHEMATICAL PRINCIPLES

[BOOK 1.

PROPOSITION XXVIII. PROBLEM XX.

To describe a trajectory giren both in kind and magnitude, given parts
of which shall be interposed between three right lines given by position.
Suppose a trajectory is to be described that

may be similar and equal to the curve line DEF,
-and may be cut by three right lines AB, AC,

BC, given by position, into parts DE and EF,

similar and equal to the given parts of this

curve line.

Draw the right lines DE, EF, DF: and

place the angles D, E, F, of this triangle DEF, so

as to touch those right lines given by position (by

Lem. XXVI). Then about the triangle describe

the trajectory, similar and equal to the curve DEF.

Q.E.F.

LEMMA XXVII.

To describe a trapezium given in kind, the angles whereof may be »,
placed, in respect of four right lines given by position, that are neither
all paralhl among themselves, nor converge to one common point, //////
the several angles may touch the several lines.
Let the four right lines ABC, AD, BD, CE, be
given by position ; the first cutting the second in A,
the third in B, and the fourth in C • and suppose a
trapezium fghi is to be described that may be similar
to the trapezium FCHI, and whose angle /, equal to
the given angle F, may touch the right line ABC ; and
(lie other angles g, h, i, equal to the other given angles,
G, H, I, may touch the other lines AD, BD, CE, re
spectively. Join FH, and upon FG. FH, FI describe J%
as many segments of circles FSG, FTH, FVI, the first
of which FSG may be capable of an angle equal to
the angle BAD ; the second FTH capable of an angle
equal to the angle CBD ; and the third FVI of an angle equal to the angle
ACE. Bnrf>, the segments are to be described towards those sides of the
lines FG, FH, FI, that the circular order of the letters FSGF may be
the same as of the letters BADB, and that the letters FTHF may turn
.ibout in the same order as the letters CBDC and the letters FVIF in the
game order as the letters ACE A. Complete the segments into entire cir
cles, and let P be the centre of the first circle FSG, Q, the centre of the
second FTH. Join and produce both ways the line PQ,, and in it take
QR in the same ratio to PQ as BC has to AB. But QR is to be taken
towards that side of the point Q that the order of the letters P, Q,, R

SEC. V.J

OF NATURAL PHILOSOPHY.

15]

may be the same as of the letters A, B, C ;
and about the centre R with the interval
RF describe a fourth circle FNc cutting
(lie third circle FVI in c. Join Fc1 cut
ting the first circle in a, and the second in
/ . Draw aG, &H, cl, and let the figure
ABC/'4f/ii be made similar to the figure
w^cFGHI; and the trapezium fghi will
be that which was required to be de
scribed.

For let the two first circles FSG, FTH
cut one the other in K ; join PK, Q,K,
RK, "K, 6K, cK, and produce QP to L.
The angles FaK, F6K, FcK at the circumferences are the halves of the
angles FPK, FQK, FRK, at the centres, and therefore equal to LPK,
LQ.K, LRK, the halves of those angles. Wherefore the figure PQRK is
iquiangular and similar to the figure «6cK, and consequently ab is to be
res PQ, to Q,R, that is, as AB to BC. But by construction, the angles
''Air, /B//,/C?', are equal to the angles F«G, F&H, Fcl. And therefore
the figure ABCfghi may be completed similar to the figure abcFGHl.
vVliich done a trapezium fghi will be constructed similar to the trapezium
FGHI, and which by its angles/, g, h, i will touch the right lines ABC,
AD, BD, CE. Q.E.F.

COR. Hence a right line may be drawn whose parts intercepted in a
given order, between four right lines given by position, shall have a given
proportion among themselves. Let the angles FGH, GHI, be so far in
creased that the right lines FG, GH, HI, may lie in directum ; and by
constructing the Problem in this case, a right line fghi will be drawn,
whose parts fg, gh, hi, intercepted between the four right lines given by
position, AB and AD, AD and BD, BD and CE, will be one to another
as the lines FG, GH, HI, and will observe the same order among them
selves. But the same thing may be more readily done in this manner.

Produce AB to K and BD to L,
so as BK may be to AB as HI to
GH ; and DL to BD as GI to FG;
and join KL meeting the right line
CE in i. Produce iL to M, so as
LM may be to iL as GH to HI ;
then draw MQ, parallel to LB, and
meeting the right line AD in g, and
join gi cutting AB, BD in f, h ; I M*
say, the thing is done.

For let MO- cut the right line AB in Q, and AD the right line KL iu

II

^52 THE MATHEMATICAL PRINCIPLES [BOOK I.

S, arid draw AP parallel to BD, and meeting iL in P, and §-M to Lh (g\
to hi, Mi to Li, GI to HI, AK to BK) and AP to BL, will be in the same
ratio. Cut DL in 11, so as DL to RL may be in that same ratio; and be
cause ffS to g~M, AS to AP. and DS to DL are proportional; therefore
(ex ceqit.o) as gS to LA, so will AS be to BL, and DS to RL ; and mixtly.
BL — RL to Lh — BL, as AS — DS to gS — AS. That is, BR is to
Eh as AD is to Ag, and therefore as BD to gQ. And alternately BR is
to BD as 13/i to g-Q,, or as fh to fg. But by construction the line BL
was cut in D and R in the same ratio as the line FI in G and H ; and
therefore BR is to BD as FH to FG. Wherefore fh is to fg as FH to
FG. Since, therefore, gi to hi likewise is as Mi to Li, that is, as GI to
HI, it is manifest that the lines FI, fi, are similarly cut in G and H, g
and //.. Q.E.F.

In the construction of this Corollary, after the line LK is drawn cutting
CE in i, we may produce iE to V, so as EV may be to Ei as FH to HI,
arid then draw V/~ parallel to BD. It will come to the same, if about the
centre i with an interval IH, we describe a circle cutting BD in X, and
produce iX to Y so as iY may be equal to IF, and then draw Yf parallel

to BO.

Sir Christopher Wren and Dr. Wallis have long ago given other solu
tions of this Problem.

PROPOSITION XXIX. PROBLEM XXI.

To describe a trajectory given in kind, that may be cut by four right
lines given by position, into parts given in order, kind, and proportion.
Suppose a trajectory is to be described that may be
similar to the curve line FGHI, and whose parts,
similar and proportional to the parts FG, GH, HI of
the other, may be intercepted between the right lines
AB and AD, AD, and BD, BD and CE given by po
sition, viz., the first between the first pair of those lines,
the second between the second, and the third between
the third. Draw the right lines FG, GH, HI, FI;
and (by Lem. XXVII) describe a trapezium fghi that
may be similar to the trapezium FGHI, and whose an
gles/, g, h, i, may touch the right lines given by posi
tion AB, AD, BD, CE, severally according to their order. And then about
bins trapezium describe a trajectory, that trajectory will be similar to the
curve line FGHI.

SCHOLIUM.

This problem may be likewise constructed in the following manner.
Joining FG, GH, HI, FI, produce GF to Y, and join FH, IG, and make

SEC. VI

OF NATURAL PHILOSOPHY.

153

El

the angles CAK. DAL equal to
the angles PGH, VFH. Let

AK, AL meet the right line
BD in K and L, and thence
draw KM, LN, of which let
KM make the angle A KM equal
to the angle CHI, and be itself
to AK as HI is to GH ; and let

LN make the angle ALN equal to the angle FHI, and be itself
to AL as HI to FH. But AK, KM. AL, LN are to be drawn
towards those sides of the lines AD, AK, AL, that the letters
OA.KMC, ALKA, DALND may be carried round in the same
order as the letters FGHIF ; and draw MN meeting the right v
line CE in L Make the angle iEP equal to the angle IGF,
and let PE be to Ei as FG to GI ; and through P draw PQ/' that may
with the right line ADE contain an angle PQE equal to the angle FIG,
and may meet the right line AB in /, and join fi. But PE and PQ arc-
to be drawn towards those sides of the lines CE, PE, that the circular
order of the letters PEtP and PEQP may be the same as of the letters
FGHIF ; and if upon the line/i, in the same order of letters, and similar
to the trapezium FGHI, a trapezium /^//.i is constructed, and a trajectory
given in kind is circumscribed about it, the Problem will be solved.

So far concerning the finding of the orbits. It remains that we deter
mine the motions of bodies in the orbits so found.

SECTION VI.

How the motions are to be found in given, orbits.

PROPOSITION XXX. PROBLEM XXII.

To find at any assigned time the place of a body moving in, a given

parabolic trajectory.

Let S be the focus, and A the principal vertex of
the parabola; and suppose 4AS X M equal to the
parabolic area to be cut off APS, which either was
described by the radius SP, since the body's departure
from the vertex, or is to be described thereby before
its arrival there. Now the quantity of that area to
be cut off is known from the time which is propor
tional to it. Bisect AS in G, and erect the perpendicular GH equal to
3M, and a circle described about th j centre H, with the interval HS, will
cut the parabola in the place P required. For letting fall PO perpendic
ular on the axis, and drawing PH, there will be AG2 -f- GH2 (=.= HP2 -_

AO^TAGJ* + PO — GH|2) = AO2 + PO2 — 2CA > — ?G!I f PO '

A G S

154 THE MATHEMATICAL PRINCIPLES [BOOK I

AG* + GH2. Whence 2GH X PO (« AO2 + PO2 — 2GAO) = AOJ

PO2
-f | PO2. For AO2 write AO X'; then dividing all the terms by

2PO; and multiplying them by 2AS, we shall have ^GH X AS (= IAO

the area APO — SPO)| = to the area APS. But GH was 3M, and

therefore ^GH X AS is 4AS X M. Wherefore the area cut off APS is
equal to the area that was to be cut off 4AS X M. Q.E.D.

Con. 1. Hence GH is to AS as the time in which the body described
the arc AP to the time in which the body described the arc between the
vertex A and the perpendicular erected from the focus S upon the axis.

COR. 2. And supposing a circle ASP perpetually to pass through the
moving body P, the xelocity of the point H is to the velocity which the
body had in the vertex A as 3 to 8; and therefore in the same ratio is
the line GH to the right line which the body, in the time of its moving
from A to P, would describe with that velocity which it had in the ver
tex A.

COR. 3. Hence, also, on the other hand, the time may be found in which
the body has described any assigned arc AP. Join AP, and on its middle
point erect a perpendicular meeting the right line GH in H,

LEMMA XXVIII.

There is no oval figure whose area, cut off by right lines at pleasure, can,
be universally found by means of equations of any number of finite
terms and dimensions.

Suppose that within the oval any point is given, about which as a pole
a right line is perpetually revolving with an uniform motion, while in
that right line a mov cable point going out from the pole moves always
forward with a velocity proportional to the square of that right line with
in the oval. By this motion that point will describe a spiral with infinite
circumgyrations. Now if a portion of the area of the oval cut off by that
right line could be found by a finite equation, the distance of the point
from the pole, which is proportional to this area, might be found by the
same equation, and therefore all the points of the spiral might be found
by a finite equation also ; and therefore the intersection of a right line
given in position with the spiral might also be found by a finite equation.
But every right line infinitely produced cuts a spiral in an infinite num
ber of points ; and the equation by which any one intersection of two lines
is found at the same time exhibits all their intersections by as many roots,
and therefore rises to as many dimensions as there are intersections. Be
cause two circles mutually cut one another in two points, one of those in-

8FC. Vl.J OF NATURAL PHILOSOPHY. 155

terscctions is not to be found but by an equation of two dimensions, fo
which the other intersection may be also found. Because there may b(-
four intersections of two conic sections, any one of them is not to be found
universally, but by an equation of four dimensions, by which they may bi>
all found together. For if those intersections are severally sought, be
cause the law and condition of all is the same, the calculus will be the
same in every case, and therefore the conclusion always the same, which
must therefore comprehend all those intersections at once within itself, and
exhibit them all indifferently. Hence it is that the intersections of the
conic se"f ions with the curves of the third order, because they may amount
to six, (\,me out together by equations of six dimensions ; and the inter
sections of two curves of the third order, because they may amount to nine,
come out together by equations of nine dimensions. If this did not ne
cessarily happen, we might reduce all solid to plane Problems, and those
higher than solid to solid Problems. But here i speak of curves irreduci
ble in power. For if the equation by which the curve is defined may bo
reduced to a lower power, the curve will not be one single curve, but com
posed of two, or more, whose intersections may be severally found by different
calculusses. After the same manner the two intersections of right lines
with the conic sections come out always by equations of two dimensions ; the
three intersections of right lines with the irreducible curves of the third
urder by equations of three dimensions ; the four intersections of right
lines with the irreducible curves of the fourth order, by equations of four
dimensions ; and so on in iitfinitum. Wherefore the innumerable inter
sections of a right line with a spiral, since this is but one simple curve
and not reducible to more curves, require equations infinite in r- .imber of
dimensions and roots, by which they may be all exhibited together. For
the law and calculus of all is the same. For if a perpendicular is let fall
from the pole upon that intersecting right line, and that perpendicular
together with the intersecting line revolves about the pole, the intersec
tions of the spiral will mutually pass the one into the other ; and that
which was first or nearest, after one revolution, will be the second ; after
two, the third ; and so on : nor will the equation in the mean time be
changed but as the magnitudes of those quantities are changed, by which
the position of the intersecting line is determined. Wherefore since those
quantities after every revolution return to their first magnitudes, the equa
tion will return to its first form ; and consequently one and the same
equation will exhibit all the intersections, and will therefore have an infi
nite number of roots, by which they may be all exhibited. And therefore
the intersection of a right line with a spiral cannot be universally found by
any finite equation ; and of consequence there is no oval figure whose area,
cut off by right lines at pleasure, can be universally exhibited by an^
such equation.

1 56

THE MATHEMATICAL PRINCIPLES

[BOOK 1

By the same argument, if the interval of the pole and point by which
the spiral is described is taken proportional to that part of the perimeter
of the oval which is cut off, it may be proved that the length of the peri
meter cannot be universally exhibited by any finite equation. But here I
speak of ovals that are not touched by conjugate figures running out in
infinitvm.

COR. Hence the area of an ellipsis, described by a radius drawn from
the focus to the moving body, is not to be found from the time given by a
finite equation ; and therefore cannot be determined by the description ol
curves geometrically rational. Those curves I call geometrically rational,
all the points whereof may be determined by lengths that are definable
by equations ; that is, by the complicated ratios of lengths. Other curves
(such as spirals, quadratrixes, and cycloids) I call geometrically irrational.
For the lengths which are or are not as number to number (according to
the tenth Book of Elements) are arithmetically rational or irrational.
And therefore I cut off an area of an ellipsis proportional to the time in
which it is described by a curve geometrically irrational, in the following
manner.

PROPOSITION XXXI. PROBLEM XXIII.

To find the place of a body moving in a given elliptic trajectory at any

assigned time.

Suppose A to be
the principal vertex,
S the focus, and O
the centre of the
ellipsis A PB ; and
let P be the place of
the body to be found.
Produce OA to G so
as OG may be to OA
as OA to OS. Erect
the perpendicular GH; and about the centre O, with the interval OG, de
scribe the circle* GEF ; and on the ruler GH, as a base, suppose the wheel
GEF to move forwards, revolving about its axis, and in the mean time by
its point A describing the cycloid ALL Which done, take GK to the
perimeter GEFG of the wheel, in the ratio of the time in which the body
proceeding from A described the arc AP, to the time of a whole revolution
in the ellipsis. Erect the perpendicular KL meeting the cycloid in L ;
then LP drawn parallel to KG will meet the ellipsis in P, the required
place of the body.

For about the centre O with the interval OA describe the semi-circle
AQB, and let LP, produced, if need be, meet the arc AQ, in Q, and join

SEC. VI.

OF NATURAL PHILOSOPHY.

157

SQ, OQ. Let OQ meet the arc EFG in F, and upon OQ let fall the
perpendicular Sll. The area APS is as the area AQS, that is, as tlie
difference between the sector OQA and the triangle OQS, or as the difLi-
ence of the rectangles *OQ, X AQ, and -J.OQ X SR, that is, because ' . >,_
is given, as the difference between the arc AQ and the right line Sll : ai.;l
therefore (because of the equality of the given ratios SR to the sine of the
arc AQ,, OS to OA, OA to OG, AQ to GF; and by division, AQ— Sii
to GF — sine of the arc AQ) as GK, the difference between the arc C 1
and tlie sine of the arc AQ. Q.E.D.

SCHOLIUM.

But since the description of this curve
is difficult, a solution by approximation
will be preferable. First, then, let there
be found a certain angle B which may
be to an angle of 57,29578 degrees,
which an arc equal to the radius subtends,
as SH, the distance of the foci, to AB,
the diameter of the ellipsis. Secondly, a certain length L, which may be to
the radius in the same ratio inversely. And these being found, the Problem
may be solved by the following analysis. By any construction (or even
by conjecture), suppose we know P the place of the body near its true
place jo. Then letting fall on the axis of the ellipsis the ordinate PR
from the proportion of the diameters of the ellipsis, the ordinate RQ of
the circumscribed circle AQB will be given ; which ordinate is the sine of
the angle AOQ, supposing AO to be the radius, and also cuts the ellipsis
in P. It will .be sufficient if that angle is found by a rude calculus in
numbers near the truth. Suppose we also know the angle proportional to
the time, that is, which is to four right a iules as the time in which tlie
body described the arc A/?, to the time of one revolution in the ellipsis.
Let this angle be N. Then take an angle D, which may be to the angle
B as the sine of the angle AOQ to the radius ; and an angle E which
may be to the angle N — AOQ -fD as the length L to the same length
L diminished by the cosine of the angle AOQ, when that angle is less
than a right angle, or increased thereby when greater. In the next
place, take an angle F that may be to the angle B as the sine of the angle
1OQ H- E to the radius, and an angle G, that may be to the angle N-
AOQ — E -f F as the length L to the same length L diminished by the
cosine of the angle AOQ + E, when that angle is less than a right angle,
or increased thereby when greater. For the third time take an angle H,
that may be to the angle B as the sine of the angle AOQ f- E 4- G to the
radius; and an angle I to the angle N — AOQ — E — G -f- H, as the

58 THE MATHEMATICAL PRINCIPLES jB(OK 1.

length L is to the same length L diminished by the cosine of the angle
AOQ -f- E + G, when that angle is less than a right angle, or increased
thereby when greater. And so we may proceed in infinitum. Lastly,
take the angle AOy equal to the angle AOQ -f- E 4- G + I -\-} &c. and
from its cosine Or and the ordinatejor, which is to its sine qr as the lesser
axis of the ellipsis to the greater, \\ e shall have p the correct place of the
body. When the angle N — AOQ, -f D happens to be negative, the
sign -|- of the angle E must be every where changed into — , and the sign —
into +. And the same thing is to be understood of the signs of the angles
G and I, when the angles N — AOQ — E -f F, and N — AOQ — E —
G + H come out negative. But the infinite series AOQ -f- E -f- G -|- I +,
&c. converges so very fast, that it will be scarcely ever needful to pro
ceed beyond the second term E. And the calculus is founded upon
this Theorem, that the area APS is as the difference between the arc
AQ and the right line let fall from the focus S perpendicularly upon the
radius OQ.

And by a calculus not unlike, the Problem
is solved in the hyperbola. Let its centre be
O, its vertex A, its focus S, and asymptote
OK ; and suppose the quantity of the area to
be cut off is known, as being proportional to
the time. Let that be A, and by conjecture
suppose we know the position of a rij;ht i ne
SP, that cuts off an area APS near the truth.
Join OP, and from A and P to the asymptote ° T A S

draw AI, PK parallel to the other asymptote ; and by the table of loga
rithms the area AIKP will be given, and equal thereto the area OPA,
which subducted from the triangle OPS, will leave the area cut off APS.
And by applying 2 APS — 2 A, or 2 A — 2 A PS, the double difference of
the area A that was to be cut off, and the area APS that is cut off, to the
line SN that is let fall from the focus S, perpendicular upon the tangent
TP, we shall have the length of the chord PQ. Which chord PQ is to
be inscribed between A and P, if the area APS that is cut off be greater
than the area A that was to be cut off, but towards the contrary side of the
point P, if otherwise : and the point Q will be the place of the body more
accurately. And by repeating the computation the place may be found
perpetually to greater and greater accuracy.

And by such computations we have a general
analytical resolution of the Problem. But the par
ticular calculus that follows is better fitted for as
tronomical purposes. Supposing AO, OB, OD, to
be the semi-axis of the ellipsis, and L its latus rec
tum, and D the difference betwixt the lesser semi-

SEC. VII.] OF NATURAL PHILOSOPHY. J 59

axis OD, and -,L the half of the latus rectum : let an angle Y be found, whose
sine may be to the radius as the rectangle under that difference J), and
AO 4- OD the half sum of the axes to the square of the greater axis AB.
Find also an angle Z, whose sine may be to the radius as the double rec
tangle under the distance of the foci SH and that difference D to triple
the square of half the greater semi-axis AO. Those angles being once
found, the place of the body may be thus determined. Take the angle T
proportional to the time in which the arc BP was described, or equal to
what is called the mean motion ; and an angle V the first equation of thr
mean motion to the angle Y, the greatest first equation, as the sine of
double the angle T is to the radius ; and an angle X, the second equation,
to the angle Z, the second greatest equation, as the cube of the sine of the
angle T is to the cube of the radius. Then take the angle BHP the mean
motion equated equal to T + X + V, the sum of the angles T, V. X,
if the angle T is less than a right angle; or equal to T + X — V, the
difference of the same, if that angle T is greater than one and less than
two right angles ; and if HP meets the ellipsis in P, draw SP, and it will
cut off the area BSP nearly proportional to the time.

This practice seems to be expeditious enough, because the angles V and
X, taken in second minutes, if you please, being very small, it will be suf
ficient to find two or three of their first figures. But it is likewise
sufficiently accurate to answer to the theory of the planet's motions.
For even in the orbit of Mars, where the greatest equation of the centre
amounts to ten degrees, the error will scarcely exceed one second. But
when the angle of the mean motion equated BHP is found, the angle oi
the true motion BSP, and the distance SP, are readily had by the known
methods.

And so far concerning the motion of bodies in curve lines. But it mav
also come to pass that a moving body shall ascend or descend in a right
line : and I shall now go on to explain what belongs to such kind of
motions.

SECTION VII.

Concerning the rectilinear ascent and descent of bodies,

PROPOSITION XXXII. PROBLEM XXIV.

Supposing that the centripetal force is reciprocally proportional to tht
square of tlie distance of the places from the centre ; it is required
to define the spaces which a body, falling directly, describes in given
times.
CASE 1. If the body does not fall perpendicularly, it will (by Cor. I

160

THE MATHEMATICAL PRINCIPLES

[BOOK I

Prop. XIII) describe some conic section whose focus is A
placed in the centre of force. Suppose that conic sec
tion to be A RPB and its focus S. And, first, if the
figure be an ellipsis, upon the greater axis thereof AB
describe the semi-circle ADB, and let the right line
I) PC pass through the falling body, making right angles
with the axis; and drawing DS, PS, the area ASD will c
be proportional to the area ASP, and therefore also to
the time. The axis AB still reaiaining the same, let the
breadth of the ellipsis be perpetually diminished, and s
the area ASD will always remain proportional to the
time. Suppose that breadth to be diminished in, in fruit um ; and the orbit
APB in that case coinciding with the axis AB, and the focus S with the
extreme point of the axis B, the body will descend in the right line AC1.
and the area ABD will become proportional to the time. Wherefore the
space AC will be given which the body describes in a given time by its-
perpendicular fall from the place A, if the area ABD is taken proportional
to the time, and from the point D the right line DC is let fall perpendic
ularly on the right line AB. Q,.E.I.

CASE 2. If the figure RPB is an hyperbola, on the
same principal diameter AB describe the rectangular
hyperbola BED ; and because the areas CSP, CB/P,
SPy 13, are severally to the sev eral areas CSD, CBED,
SDEB, in the given ratio of the heights CP, CD, and
the area SP/B is proportional to the time in which
the body P will move through the arc P/B. the area
SDEB will be also proportional to that time. Let
the latus rectum of the hyperbola RPB be diminished
in infitiitum, the latus transversum remaining the
same; and the arc PB will come to coincide with the
right line CB, and the focus S, with the vertex B, A-
and the right line SD with the right line BD. And therefore the area
BDEB will be proportional to the time in which the body C, by its per
pendicular descent, describes the line CB. Q.E.I.

CASE 3. And by the like argument, if the figure
RPB is a parabola, and to the same principal ver
tex B another parabola BED is described, that
may always remain given while the former para
bola in whose perimeter the body P moves, by
having its latus rectum diminished and reduced
to nothing, comes to coincide with the line CB,
the parabolic segment BDEB will be proportional if
to the time in which that body P or C will descend to the centre S or B
Q.K.T

fl.l

OF NATURAL PHILOSOPHY.

PROPOSITION XXXIII. THEOREM IX.

The tilings above found being supposed. I say, thai the, velocity of a Jai
ling body in any place C is to the velocity of a body, describing a
circle about the centre B at the distance BC, in, the subduplicate ratio
of AC, the distance of the body from the remoter vertex A of the circle
or rectangular hyperbola, to iAB, the principal semi-diameter of the

Let AB, the common dia
meter of both figures RPB,
DEB, be bisected in O; and
draw the right line PT that
may touch the figure RPB
in P, and likewise cut that
common diameter AB (pro
duced, if need be) in T; and
let SY be perpendicular to
this line, and BQ to this di
ameter, and suppose the latus
rectum of the figure RPB to
be L. Prom Cor. 9, Prop.
XVI, it is manifest that the
velocity of a body, moving
in the line RPB about the
centre S, in any place P, is
to the velocity of a body describing a circle about the same centre, at the
distance SP, in the subduplicate ratio of the rectangle £L X SP to SY2
Por by the properties of the conic sections ACB is to CP2 as 2AO to L.

2CP5 X AO

and therefore — — rrrr; is equal to L. Therefore those, velocities an

o--

ACB

to each other in the subduplicate ratio of

CP3 X AO X SP

ACB

toSY~. More

over, by the properties of the conic sections, CO is to BO as BO to TV.?
and (by composition or division) as CB to BT. Whence (by division cs
composition) BO — or + CO will be to BO as CT to BT, that is, AC

CP2 X AO X SP

ACB"

is equal to

will be to AO as CP to BQ; and therefore

~AO X BC * ^°W suPPose GV, tne breadth of the figure RPB, to

be diminished in infinitum, so as the point P may come to coincide with
the point C, and the point S with the point B. and the line SP with the
line BC, and the line SY with the line BQ; and the velocity of the body
now descending perpendicularly in the line CB will be to the velocity of

11

162

THE MATHEMATICAL PRINCIPLES

[BOOK 1

a body describing a circle about the centre B, at the distance BC, in thr

BQ2 X AC X SP

subduplicate ratio of - -r-^ - ^ - to SY2, that is (neglecting the ra-

X Jo

tios of equality of SP to BC, and BQ,2 to SY2), in the subduplicate ratio
of AC to AO, or iAB. Q.E.D.

COR. 1 . When the points B and S come to coincide, TC will become to
TS as AC to AO.

COR. 2. A body revolving in any circle at a given distance from the
centre, by its motion converted upwards, will ascend to double its distance

from the centre.

•

PROPOSITION XXXIV. THEOREM X.

If the. figure BED is a parabola, I say, that the velocity of a falling
body in any place C is equal to the velocity by which a body may
uniformly describe a circle about the centre B at half the interval BC
For (by Cor. 7, Prop. XVI) the velocity of a

body describing a parabola RPB about the cen

tre S, in any place P, is equal to the velocity of

a body uniformly describing a circle about the c

same centre S at half the interval SP. Let the

breadth CP of the parabola be diminished in

itifiiiitirni, so as the parabolic arc P/B may come

to coincide with the right line CB, the centre S s

with the vertex B, and the interval SP with the

interval BC, and the proposition will be manifest. Q.E.D.

PROPOSITION XXXV. THEOREM XL

The same things supposed, I say, that the area of the figure DES, de
scribed by the indefinite radius SD, is equal to the area which a body
with a radius equal to h'df the latus rectum of the figure DES, by
uniformly revolving about the centre S, may describe in the same tijiw.

1

JD/

AJ

SEC. ni:

OF NATURAL PHILOSOPHY.

For suppose a body C in the smallest moment of time describes in fal
ling the infinitely little line Cc. while another body K, uniformly revolv
ing about the centre S in the circle OK/r, describes the arc KA:. Erect the
perpendiculars CD, cd, meeting the figure DES in D, d. Join SD, Sf/.
SK. SA*; and draw Del meeting the axis AS in T, and thereon let fall the
perpendicular SY.

CASE 1. If the figure DES is a circle, or a rectangular hyperbola, bisect
its transverse diameter AS in O, and SO will be half the latus rectum.
And because TC is to TD as Cc to Dd, and TD to TS as CD to S Y ;
ex aquo TC will be to TS as CD X Cc to SY X Dd. But (by Cor. 1,
Prop. XXXIII) TC is to TS as AC to AO; to wit, if in the coalescence
of the points D, d, the ultimate ratios of the lines are taken. Wherefore
AC is to AO or SK as CD X Cc to S Y X Vd, Farther, the velocity of
the descending body in C IF, to the velocity of a body describing a circle
about the centre S, at the interval SC, in the subduplicate ratio of AC to
AO or SK (by Pi-op. XXXIII) ; and this velocity is to the velocity of a
body describing the circle OKA: in the subduplicate ratio of SK to SC
(by Cor. 6, Prop IV) ; and, ex aqnnj the first velocity to the last, that is,
the little line Cc to the arc K/r, in the subduplicate ratio of AC to SC,
that is, in the ratio of AC to CD. Wherefore CD X Cc is equal to AC
X KA*, and consequently AC to SK as AC X KA: to SY X IW. and
thence SK X KA: equal to SY X Drf, and iSK X KA: equal to £SY X DC/,
that is, the area KSA* equal to the area SDrf. Therefore in every moment
of time two equal particles, KSA" and SDrf, of areas are generated, which,
if their magnitude is diminished, and their number increased in iiifinif't-w,
obtain the ratio cf equality, and consequently (by Cor. Lem. IV), the whole
areas together generated are always equal. Q..E.D.

CASE 2. But if the figure DES is a
parabola, we shall find, as above. CD X
Cc to SY X Df/ as TC to TS, that is,
as 2 to 1 ; and that therefore |CD X Cc
is equal to i SY X Vd. But the veloc
ity of the falling body in C is equal to
the velocity writh which a circle may be
uniformly described at the interval 4SC
(by Prop" XXXIV). And this velocity
to the velocity with which a circle may
be described with the radius SK, that is,
the little line Cc to the arc KA', is (by
Cor. 6, Prop. IV) in the subduplicate ratio of SK to iSC ; that is, in the
ratio of SK to *CD. Wherefore iSK X KA: is equal to 4CD X Cc, and
therefore equal to £SY X T)d ; that is, the area KSA* is equal to the area
SIW, as above. Q.E.D.

164

THE MATHEMATICAL PRINCIPLES

[BOOK 1.

PROPOSITION XXXVI. PROBLEM XXV.

To determine the times of the descent of a body falling from

place A.

Upon the diameter AS, the distance of the body from the
centre at the beginning, describe the semi-circle ADS, as
likewise the semi-circle OKH equal thereto, about the centre
S. From any place C of the body erect the ordinate CD. °
Join SD, and make the sector OSK equal to the area ASD.
It is evident (by Prop. XXXV) that the body in falling will
describe the space AC in the same time in which another body,
uniformly revolving about the centre S, may describe the arc
OK. Q.E.F. M

a given

PROPOSITION XXXVII. PROBLEM XXVI.

To define the times of the ascent or descent of a body projected upwards

or downwards from a given place.

Suppose the body to go oif from the given place G, in the direction of
the line GS, with any velocity. In the duplicate ratio of this velocity to
the uniform velocity in a circle, with which the body may revolve about

\

H

D

the centre S at the given interval SG, take GA to £AS. If that ratio is
the same as of the number 2 to 1, the point A is infinitely remote ; in
which case a parabola is to be described with any latus rectum to the ver
tex S, and axis SG ; as appears by Prop. XXXIV. But if that ratio is
less or greater than the ratio of 2 to 1, in the former case a circle, in the
latter a rectangular hyperbola, is to be described on the diameter SA; as
appears by Prop. XXXIII. Then about the centre S, with an interval
equal to half the latus rectum, describe the circle H/vK ; and at the place
G of the ascending or descending body, and at any other place C, erect the
perpendiculars GI, CD, meeting the conic section or circle in I and D.
Then joining SI, SD, let the sectors HSK, HS& be made equal to the
segments SEIS, SEDS. and (by Prop. XXXV) the body G will describe

SEC. VII.]

OF NATURAL PHILOSOPHY.

165

the space GO in the same time in which the body K may describe t*he arc
Kk. Q.E.F.

PROPOSITION XXXVIII. THEOREM XII.

Supposing that the centripetal force is proportional to the altitude or
distance of places from the centre, I say, that the times and velocities
of falling bodies, and the spaces which they describe, are respectively
proportional to the arcs, and the right and versed sines of the arcs.
Suppose the body to fall from any place A in the A.
right line AS ; and about the centre of force S, with
the interval AS, describe the quadrant of a circle AE ;
and let CD be the right sine of any arc AD ; and the
body A will in the time AD in falling describe the
space AC, and in the place C will acquire the ve
locity CD.

This is demonstrated the same way from Prop. X, as Prop. XXX11 was
demonstrated from Prop. XI.

COR. 1. Hence the times are equal in which one body falling from the
place A arrives at the centre S, and another body revolving describes the
quadrantal arc ADE.

COR. 2. Wherefore all the times are equal in which bodies falling from
whatsoever places arrive at the centre. For all the periodic times of re
volving bodies are equal (by Cor. 3; Prop. IV).

PROPOSITION XXXIX. PROBLEM XXVIT.

Supposing a centripetal force of any kind, and granting the quadra-
tnres of curvilinear figures ; it is required to find the velocity of a bod)/,
ascending or descending in a right line, in the several places through
which it passes ; as also the time in which it will arrive at any place :
and vice versa.
Suppose the body E to fall from any place

A in the right line AD EC ; and from its place

E imagine a perpendicular EG always erected

proportional to the centripetal force in that

place tending to the centre C ; and let BFG

be a curve line, the locus of the point G. And D

in the beginning of the motion suppose EG to

coincide with the perpendicular AB ; and the

velocity of the body in any place E will be as

a right line whose square is equal to the cur
vilinear area ABGE. Q.E.I.
In EG take EM reciprocally proportional to

E

366 THE MATHEMATICAL PRINCIPLES [BOOK 1

a right line whose square is equal to the area ABGE, and let VLM be a
curve line wherein the point M is always placed, and to which the right
line AB produced is an asymptote; and the time in which the body in
falling- describes the line AE, will be as the curvilinear area ABTVME.
Q.E.I.

For in the right line AE let there be taken the very small line DE of
a given length, and let DLF be the place of the line EMG, when the
body was in D ; and if the centripetal force be such, that a right line,
whose square is equal to the area ABGE; is as the velocity of the descend
ing body, the area itself will be as the square of that velocity ; that is, if
for the velocities in D and E we write V and V + I, the area ABFD will
be as VY, and the area ABGE as YY + 2VI -f II; and by division, the

area DFGE as 2 VI -f LI, and therefore ^ will be as -- ^r

that is. if we take the first ratios of those quantities when just nascent, the

2YI

length DF is as the quantity -|yrr» an(i therefore also as half that quantity

1 X Y

• But the time in which the body in falling describes the very

line DE, is as that line directly and the velocity Y inversely ; and
the force will be as the increment I of the velocity directly and the time
inversely ; and therefore if we take the first ratios when those quantities

I X V

are just nascent, as— -jy==r-. that is, as the length DF. Therefore a force

proportional to DF or EG will cause the body to descend with a velocity
that is as the right line whose square is equal to the area ABGE. Q.E.D.

Moreover, since the time in which a very small line DE of a given
length may be described is as the velocity inversely, and therefore also
inversely as a right line whose square is equal to the area ABFD ; and
since the line DL. and by consequence the nascent area DLME, will be as
(he same right line inversely, the time will be as the area DLME, and
the sum of all the times will be as the sum of all the areas : that is (by
Cor. Lern. IV), the whole time in which the line AE is described will be
as the whole area ATYME. Q.E.D.

COR. 1. Let P be the place from whence a body ought to fall, so as
that, when urged by any known uniform centripetal force (such as
gravity is vulgarly supposed to be), it may acquire in the place D a
velocity equal to the velocity which another body, falling by any force
whatever, hath acquired in that place D. In the perpendicular DF let
there be taken DR., which may be o DF as that uniform force to
the other force in the place D. Complete the rectangle PDRQ,, and cut
iff the area. ABFD equal to that rectangle. Then A will be the place

SEC. VII. I

OF NATURAL PHILOSOPHY.

10;

from whence the other body fell. For com
pleting the rectangle DRSE, since the area
ABFD is to the area DFGE as VV to 2VI,
and therefore as 4V to I, that is, as half the
whole velocity to the increment of the velocity
of the body falling by the unequable force ; and
in like manner the area PQRD to the area
DRSE as half the whole velocity to the incre
ment of the velocity of the body falling by the
uniform force ; and since those increments (by
reason of the equality of the nascent times)
are as the generating forces, that is, as the or-
dinates DF, DR, and consequently as the nascent areas DFGE, DRSE :
therefore, ex aq-uo, the whole areas ABFD, PQRD will be to one another
as the halves of the whole velocities ; and therefore, because the velocities
are equal, they become equal also.

COR. 2. Whence if any body be projected either upwards or downwards
with a given velocity from any place D, and there be given the law of
centripetal force acting on it, its velocity will be found in any other place,
as e, by erecting the ordinate eg, and taking that velocity to the velocity
in the place D as a right line whose square is equal to the rectangle
PQRD, either increased by the curvilinear area DFge, if the place e is
below the place D, or diminished by the same area DFg-e, if it be higher,
is to the right line whose square is equal to the rectangle PQRD alone.

COR. 3. The time is also known by erecting the ordinate em recipro
cally proportional to the square root of PQRD -f- or — T)Fge, and taking
the time in which the body has described the line De to the time in which
another body has fallen with an uniform force from P, and in falling ar
rived at D in the proportion of the curvilinear area DLme to the rectan
gle 2PD X DL. For the time in which a body falling with an uniform
force hath described the line PD, is to the time in which the same body
has described the line PE in the subduplicate ratio of PD to PE ; that is
(the very small line DE being just nascent), in the ratio of PD to PD -f
^DE; or 2PD to 2PD -f- DE, and, by division, to the time in which the
body hath described the small line DE, as 2PD to DE, and therefore as
the rectangle 2PD X DL to the area DLME ; and the time in which
both the bodies described the very small line DE is to the time in which
the body moving unequably hath described the line De as the area DLME
to the area DLme ; and, ex aquo, the first mentioned of these times is to
the last as the rectangle 2PD X DL to the area DLrae.

163 THE MATHEMATICAL PRINCIPLES [BoOK I

SECTION VIII.

Of the invention of orbits wherein bodies will revolve, being acted upon
by any sort of centripetal force.

PROPOSITION XL. THEOREM XIII.

//' a body, acted upon by any centripetal force, is any how moved, and
another body ascends or descends in a right line, and their velocities
be equal in amj one case of equal altitudes, t/ieir velocities will be also
equal at all equal altitudes.

Let a body descend from A through D and E, to the centre
(j : and let another body move from V in the curve line VIK&.
From the centre C, with any distances, describe the concentric
circles DI, EK, meeting the right line AC in I) and E; and
the curve VIK in I and K. Draw 1C meeting KE in N, and
on IK let fall the perpendicular NT • and let the interval DE
or IN between the circumferences of the circles be very small ; K/
and imagine the bodies in D and I to have equal velocities.
Then because the distances CD and CI are equal, the centri
petal forces in D and I will be also equal. Let those forces be k)
expressed by the equal lineoke DE and IN ; and let the force
IN (by Cor. 2 of the Laws of Motion) be resolved into two
others, NT and IT. rl hen the force NT acting in the direction
line NT perpendicular to the path ITK of the body will not at all affect
or change the velocity of the body in that path, but only draw it aside
from a rectilinear course, and make it deflect perpetually from the tangent
of the orbit, and proceed in the curvilinear path ITK/j. That whole
force, therefore, will be spent in producing this effect: but the other force
IT, acting in the direction of the course of the body, will be all employed
in accelerating it, and in the least given time will produce an acceleration
proportional to itself. Therefore the accelerations of the bodies in D and
I, produced in equal times, are as the lines DE, IT (if we take the first
ratios of the nascent lines DE, IN, IK, IT, NT) ; and in unequal times as
those lines and the times conjunctly. But the times in which DE and IK
are described, are, by reason of the equal velocities (in D and I) as the
spaces described DE and IK, and therefore the accelerations in the course
of the bodies through the lines DE and IK are as DE and IT, and DE
and IK conjunctly ; that is, as the square of DE to the rectangle IT into
IK. But the rectangle IT X IK is equal to the square of IN, that is,
equal to the square of DE ; and therefore the accelerations generated in
the passage of the bodies from D and I to E and K are equal. Therefore
the velocities of the holies in E and K are also equal, and by the same
reasoning they will always be found equal in any subsequent equal dis
tances. Q..E.D.

SEC. VI11.J OF NATURAL PHILOSOPHY. 169

By the same reasoning, bodies of equal velocities and equal distances
from the centre will be equally retarded in their ascent to equal distances.
Q.E.D.

COR. 1. Therefore if a body either oscillates by hanging to a string, or
by any polished and perfectly smooth impediment is forced to move in a
curve line ; and another body ascends or descends in a right line, and their
velocities be equal at any one equal altitude, their velocities will be also
equal at all other equal altitudes. For by the string of the pendulous
body, or by the impediment of a vessel perfectly smooth, the same thing
will be effected as by the transverse force NT. The body is neither
accelerated nor retarded by it, but only is obliged to leave its rectilinear
course.

COR. 2. Suppose the quantity P to be the greatest distance from the
centre to which a body can ascend, whether it be oscillating, or revolving
in a trajectory, and so the same projected upwards from any point of a
trajectory with the velocity it has in that point. Let the quantity A be
the distance of the body from the centre in any other point of the orbit ; and
let the centripetal force be always as the power An — ', of the quantity A, the
index of which power n — 1 is any number n diminished by unity. Then
the velocity in every altitude A will be as v/ P11 — A", and therefore will
be given. For by Prop. XXXIX, the velocity of a body ascending and
descending in a right line is in tha't very ratio.

PROPOSITION XLI. PROBLEM XXVTII.

Supposing a centripetal force of any kind, and granting the quadra
tures of curvilinear figures, it is required to find as well the trajecto
ries in which bodies will move, as the times of their motions in the
trajectories found.
Let any centripetal force tend to
the centre C, and let it be required
to find the trajectory VIKAr. Let R,
there be given the circle VR, described
from the centre C with any interval
CV; and from the same centre de
scribe any other circles ID, KE cut
ting the trajectory in I and K, and
the right line CV in D and E. Then
draw the right line CNIX cutting the c

circles KE, VR in N and X, and the right line CKY meeting the circle
VJi in Y. Let the points I and K be indefinitely near ; and let the body
go on from V through I and K to k ; and let the point A be the place
from whence anothe body is to fall, so as in the place D to acquire a ve
locity equal to the velocity of the first body in I. And things remaining
as in Prop. XXXIX, the lineola IK, described in the least given time

THE MATHEMATICAL PRINCIPLES [BOOK 1

trill be as the velocity, and therefore as the right line whose square is
equal to the area ABFD, and the triangle ICK proportional to the time
will be given, and therefore KN will be reciprocally as the altitude 1C :
that is (if there be given any quantity Q, and the altitude 1C be called

A), as -T-. This quantity — call Z, and suppose the magnitude of Q, to

oe such that in some case v/ABFD may be to Z as IK to KN, and then
in all cases V ABFD will be to Z as IK to KN, and ABFD to ZZ as
IK2 to KN2, and by division ABFD — ZZ to ZZ as IN2 to KN2, and there-

fore V ABFD — ZZ to Z, or — as IN to KN; and therefore A x KN

Q. x IN

\vill be equal to — — — . Therefore since YX X XC is to A X KN

— ZZ

Q. X IN x CX2
as CX2, to AA, the rectangle XY X XC will be equal to-

AAv/ABFD — ZZ.

Therefore in the perpendicular DF let there be taken continually I)//, IV

Q ax ex2

equal to — , =. — respectively, and

2 v/ ABFD — ZZ 2AA V ABFD — ZZ

let the curve lines ab, ac, the foci of the points b and c, be described : and
from the point V let the perpendicular Va be erected to the line AC, cut
ting off the curvilinear areas VD&a, VDra, and let the ordi nates Es:?
E#, be erected also. Then because the rectangle D& X IN or DbzR is
equal to half the rectangle A X KN, or to the triangle ICK ; and the
rectangle DC X IN or Dc.rE is equal to half the rectangle YX X XC, or
to the triangle XCY; that is, because the nascent particles I)6d3, ICK
of the areas VD/>#, VIC are always equal; and the nascent particles
Dc^-E, XCY of the areas VDca, VCX are always equal : therefore the
generated area VD6a will be equal to the generated area VIC, and there
fore proportional to the time; and the generated area VDco- is equal to
the generated sector VCX. If, therefore, any time be given during which
the body has been moving from V, there will be also given the area pro
portional to it VD/>«; and thence will be given the altitude of the body
CD or CI ; and the area VDca, and the sector VCX equal there' o, together
with its angle VCL But the angb VCI, and the altitude CI being given,
there is also given the place I, in which the body will be found at the end
of that time. Q.E.I.

COR. 1. Hence the greatest and least altitudes of the bodies, that is, the
apsides of the trajectories, may be found very readily. For the apsides
are those points in which a right line 1C drawn through the centre falls
perpendicularly upon the trajectory VTK; which comes to pass when the
right lines IK and NK become equal; that is, when the area ABFD ig
C'nl to ZZ.

OF NATURAL PHILOSOPHY.

171

SEC. VI1LJ

COR. 2. So also the angle KIN, in which the trajectory at any place
cuts the line 1C. may be readily found by the given altitude 1C of the
body : to wit, by making the sine of that angle to radius as KN to IK
that is, as Z to the square root of the area ABFD.

COR. 3. If to the centre C, and the
principal vertex V, there be described a
conic section VRS ; and from any point
thereof, as R, there be drawn the tangent T
RT meeting the axis CV indefinitely pro
duced in the point T ; and then joining C
CR there be drawn the right line CP, Q-
equal to the abscissa CT, making an angle VCP proportional to the sector
VCR ; and if a centripetal force, reciprocally proportional to the cubes
of the distances of the places from the centre, tends to the centre C ; and
from the place V there sets out a body with a just velocity in the direc
tion of a line perpendicular to the right, line CV; that body will proceed
in a trajectory VPQ,, which the point P will always touch ; and therefore
if the conic section VI\ S be an hyberbola, the body will descend to the cen
tre ; but if it be an ellipsis, it will ascend perpetually, and go farther and
farther off in infinilum. And, on the contrary, if a body endued with any
velocity goes off from the place V, and according as it begins either to de
scend obliquely to the centre, or ascends obliquely from it, the figure VRS
be either an hyperbola or an ellipsis, the trajectory may be found by increas
ing or diminishing the angle VCP in a given ratio. And the centripetal
force becoming centrifugal, the body will ascend obliquely in the trajectory
VPQ, which is found by taking the angle VCP proportional to the elliptic
sector VRC, and the length CP equal to the length CT, as before. All these
things follow from the foregoing Proposition, by the quadrature of a certain
ourve, the invention of which, as being easy enough, for brevity's sake I omit.

PROPOSITION XLII. PROBLEM XXIX.

The law of centripetal force being given, it is required to find the motion
of a body setting out from a given place, with a given velocity, in the
direction of a given right line.
Suppose the same things as in

Ihe three preceding propositions;

and let the body go off from

the place I in the direction of the

little line, IK, with the same ve
locity as another body, by falling

with an uniform centripetal force

from the place P, may acquire in

I); and let this uniform force be

to the force with which the body

1.72 THE MATHEMATICAL PRINCIPLES [BOOK 1.

is at first urged in I, as DR to DF. Let the body go on towards k; and
about the centre C, with the interval Ck, describe the circle ke, meeting
the right line PD in e, and let there be erected the lines eg, ev, ew, ordi-
nately applied to the curves BF§*, abv} acw. From the given rectangle
PDRQ, and the given law of centripetal force, by which the first body is
acted on, the curve line BF»* is also given, by the construction of Prop.
XXVII, and its Cor. 1. Then from the given angle CIK is given the
proportion of the nascent lines 1K; KN ; and thence, by the construction
of Prob. XXVIII, there is given the quantity Q,, with the curve lines abv,
acw ; and therefore, at the end of any time Dbve, there is given both
the altitude of the body Ce or Ck, and the area Dcwe, with the sector
equal to it XCy, the angle 1C A:, and the place k} in which the body will
then be found. Q.E.I.

We suppose in these Propositions the centripetal force to vary in its
recess from the centre according to some law, which any one may imagine
at pleasure; but at equal distances from the centre to be everywhere the
Bame.

I have hitherto considered the motions of bodies in immovable orbits.
It remains now to add something concerning their motions in orbits which
revolve round the centres of force.

SECTION IX.

Of the motion of bodies in moveable orbits ; and of the motion of the

apsides.

PROPOSITION XLIII. PROBLEM XXX.

Ft is required to make a body move in a trajectory that revolves about
the centre of force in the same manner as another body in the same
trajectory at rest.
In. the orbit VPK, given by position, let the body

P revolve, proceeding from V towards K. From

the centre C let there be continually drawn Cp, equal

to CP, making the angle VC/? proportional to the

angle VCP ; and the area which the line Cp describes

will be to the area VCP, which the line CP describes

at the same time, ns the velocity of the describing

line Cp to the velocity of the describing line CP ;

that is, as the angle VC/? to the angle VCP, therefore in a given ratio,

and therefore proportional to the time. Since, then, the area described by

the line Cp in an immovable plane is proportional to the time, it is manifest

that a body, being acted upon by a just quantity of centripetal force may

SEC. L\.]

OF NATURAL PHILOSOPHY.

173

revolve with the point p in the curve line which the same point p, by the
method just now explained, may be made to describe an immovable plane.
Make the angle VC^ equal to the angle PC/?, and the line Cu equal to
CV, and the figure uCp equal to the figure VCP; and the body being al
ways in the point p} will move in the perimeter of the revolving figure
nCp, and will describe its (revolving) arc up in the same time the* the
other body P describes the similar and equal arc VP in the quiescov.t fig
ure YPK. Find, then, by Cor. 5, Prop. VI., the centripetal force by which
the body may be made to revolve in the curve line which the pom* p de
scribes in an immovable plane, and the Problem will be solved. O/E.K.

PROPOSITION XLIV. THEOREM XIV.

The difference of the forces, by which two bodies may be madi, to KMVG
equally, one in a quiescent, the other in the same orbit revolving, i 1 in
a triplicate ratio of their common altitudes inversely.
Let the parts of the quiescent or
bit VP, PK be similar and equal to
the parts of the revolving orbit up,
pk ; and let the distance of the points
P and K be supposed of the utmost
smallness Let fall a perpendicular
kr from the point k to the right line
pC, and produce it to m, so that mr
may be to kr as the angle VC/? to the /2\-
angle VCP. Because the altitudes
of the bodies PC and pV, KG and
kC} are always equal, it is manifest
that the increments or decrements of
the lines PC and pC are always
equal ; and therefore if each of the
several motions of the bodies in the places P and p be resolved into two
(by Cor. 2 of the Laws of Motion), one of which is directed towards the
centre, or according to the lines PC, pC, and the other, transverse to the
former, hath a direction perpendicular to the lines PC and pC ; the mo
tions towards the centre will be equal, and the transverse motion of the
body p will be to the transverse motion of the body P as the angular mo
tion of the line pC to the angular motion of the line PC ; that is, as the
angle VC/? to the angle VCP. Therefore, at the same time that the bodv
P, by both its motions, comes to the point K, the body p, having an equal
motion towards the centre, will be equally moved from p towards C ; arid
therefore that time being expired, it will be found somewhere in the
line mkr, which, passing through the point k, is perpendicular to the line
pC ; and by its transverse motion will acquire a distance from the line

174 THE MATHEMATICAL PRINCIPLES [BOOK J.

»C, that will be to the distance which the other body P acquires from the
line PC as the transverse motion of the body p to the transverse motion of
the other body P. Therefore since kr is equal to the distance which the
body P acquires from the line PC, and mr is to kr as the angle VC/? to
the angle VCP, that is, as the transverse motion of the body p to the
transverse motion of the body P, it is manifest that the body p, at the ex
piration of that time, will be found in the place m. These things will be
so, if the bodies jo and P are equally moved in the directions of the lines
pC and PC, and are therefore urged with equal forces in those directions.
I: ut if we take an angle pCn that is to the angle pCk as the angle VGj0
to the angle VCP, and nC be equal to kG, in that case the body p at the
expiration of the time will really be in n ; and is therefore urged with a
greater force than the body P, if the angle nCp is greater than the angle
kCp, that is, if the orbit npk, move either in cmiseqnentia, or in antece-
denticij with a celerity greater than the double of that with which the line
CP moves in conseqnentia ; and with a less force if the orbit moves slower
in antecedent-la. And ihj difference of the forces will be as the interval
mn of the places through which the body would be carried by the action of
that difference in that given space of time. About the centre C with the
interval Cn or Ck suppose a circle described cutting the lines mr, tun pro
duced in s and £, and the rectangle mn X nit will be equal to the rectan-

*//? n ^* */?? ^

•"•le mk X ins, and therefore mn will be equal to — — . But since

mt

the triangles pCk, pCn, in a given time, are of a given magnitude, kr and
mr. a id their difference mk, and their sum ms, are reciprocally as the al
titude pC, and therefore the rectangle mk X ms is reciprocally as the
square of the altitude pC. But, moreover, mt is directly as |//z/, that is, as
the altitude pC. These are the first ratios of the nascent lines ; and hence

— r — - that is, the nascent lineola mn. and the difference of the forces

mt

proportional thereto, are reciprocally as the cube of the altitude pC.
Q.E.D.

COR. I. Hence the difference of the forces in the places P and p, or K and
/.*, is to the force with which a body may revolve with a circular motion
from R to K, in the same time that the body P in an immovable orb de
scribes the arc PK, as the nascent line m,n to the versed sine of the nascent

mk X ms rk2

arc RK, that is, as — — to ^g, or as mk X ms to the square of

rk ; that is. if we take given quantities F and G in the same ratio to one
another as the angle VCP bears to the angle VQ?, as GG — FF to FF.
And, therefore, if from the centre C, with any distance CP or Cp, there be
described a circular sector equal to the whole area VPC, which the body

OEC.

IX.l

OF NATURAL PHILOSOPHY.

175

revolving in an immovable orbit has by a radius drawn to the centre de-
bribed in any certain time, the difference of the forces, with which the
body P revolves in an immovable orbit, and the body p in a movable or
bit, will be to the centripetal force, with which another body by a radius
drawn to the centre can uniformly describe that sector in the same time
as the area VPC is described, as GG — FF to FF. For that sector and
the area pCk are to one another as the times in which they are described.

COR. 2. If the orbit YPK be an
ellipsis, having its focus C, and its
highest apsis Y, and we suppose the
the ellipsis upk similar and equal to ..
it, so that pC may be always equal /
to PC, and the angle YC/? be to the •;
angle YCP in the given ratio of G \
to F ; and for the altitude PC or pC \
we put A, and 2R for the latus rec- /t\
turn of the ellipsis, the force with *
which a body may be made to re
volve in a movable ellipsis will be as

FF RGG — RFF

— - + - — -rg — — , and vice versa.
/Y A. A.

Let the force with which a body may

revolve in an immovable ellipsis be expressed by the quantity , and the

-. 7

force in V will be

FF

But the force with which a body may revolve in

a circle at the distance CY, with the same velocity as a body revolving in
an ellipsis has in Y, is to the force with which a body revolving in an ellip
sis is acted upon in the apsis Y, as half the latus rectum of the ellipsis to the

RFF

semi-diameter CY of the circle, and therefore is as , =- : and tlu

RFF

which is to this, as GG — FF to FF, is as - ~py^~~ ~: and this force

(by Cor. 1 cf this Prop.) is the difference of the forces in Y, with which the
body P revolves in the immovable ellipsis YPK, and the body p in the
movable ellipsis upk. Therefore since by this Prop, that difference at

any other altitude A is to itself at the altitude CY as -r-, to ^TF-» the same

AJ CYJ

R C^ ( "* R P^ T*1

difference in every altitude A will be as - — 3 — : — . Therefore to the

FF
force -T-: , by which the body may revolve in an immovable ellipsis VPK

176 THE MATHEMATICAL PRINCIPLES [BOOK I.

idd the excess — -:-= , and the sum will be the whole force -r-r -\-

A AA

RGG — RFF,

.-5 — by which a body may revolve in the same time in the mot-

A.

•able ellipsis upk.

COR. 3. In the same manner it will be found, that, if the immovable or
bit VPK be an ellipsis having its centre in the centre of the forces C} and
there be supposed a movable ellipsis -upk, similar, equal, and concentrical
to it ; and 2R be the principal latus rectum of that ellipsis, and 2T the
latus transversum, or greater axis ; and the angle VCjo be continually to the
angle TCP as G to F ; the forces with which bodies may revolve in the im-

FFA FFA

movable and movable ellipsis, in equal times, will be as ^ and -p™~

RGG — RFF

+ — .-3 respectively.

A

COR. 4. And universally, if the greatest altitude CV of the body be called
T, and the radius of the curvature which the orbit VPK has in Y, that is,
the radius of a circle equally curve, be called R, and the centripetal force
with which a body may revolve in any immovable trajectory VPK at the place

VFF
V be called -f-=Trri , and in other places P be indefinitely styled X ; and the

altitude CP be called A, and G be taken to F in the given ratio of the
angle VCjD to the angle VCP ; the centripetal force with which the same
body will perform the same motions in the same time, in the same trajectory
•upk revolving with a circular motion, will be as the sum of the forces X -f-
VRGG — VRFF

~ A*

COR. 5. Therefore the motion of a body in an immovable orbit being
given, its angular motion round the centre of the forces may be increased
or diminished in a given ratio; and thence new immovable orbits may be
found in which bodies may revolve with new centripetal forces.

COR. 6. Therefore if there be erected the line VP of an indeterminate
-p length, perpendicular to the line CV given by po

sition, and CP be drawn, and Cp equal to it, mak
ing the angle VC/? having a given ratio to the an
gle VCP, the force with which a body may revolve
in the curve line Vjo/r, which the point p is con
tinually describing, will be reciprocally as the cube
C of the altitude Cp. For the body P, by its vis in
ertia alone, no other force impelling it, will proceed uniformly in the right
line VP. Add, then, a force tending to the centre C reciprocally as the
cube of the altitude CP or Cp, and (by what was just demonstrated) the

SEC. IX..J OF NATURAL PHILOSOPHY. 177

body will deflect from the rectilinear motion into the curve line Ypk. But
this curve ~Vpk is the same with the curve VPQ found in Cor. 3, Prop
XLI, in which, I said, hodies attracted with such forces would ascend
obliquely.

PROPOSITION XLV. PROBLEM XXXL

To find the motion of the apsides in orbits approaching very near to

circles.

This problem is solved arithmetically by reducing the orbit, which a
body revolving in a movable ellipsis (as in Cor. 2 and 3 of the above
Prop.) describes in an immovable plane, to the figure of the orbit whose
apsides are required ; and then seeking the apsides of the orbit which that
body describes in an immovable plane. But orbits acquire the same figure,
if the centripetal forces with which they are described, compared between
themselves, are made proportional at equal altitudes. Let the point V be
the highest apsis, and write T for the greatest altitude CV, A for any other
altitude CP or C/?, and X for the difference of the altitudes C V — CP :
and the force writh which a body moves in an ellipsis revolving about its

•p -p T? C* f ^ T? F* F

focus C (as in Cor. 2), and which in Cor. 2 was as -r-r -\ — — -.-3 — — ,

FFA + RGG — RFF ,

that is as, -^ — , by substituting T — X for A, will be-

A

RGG — RFF + TFF — FFX

come as -p — . In like manner any other cen
tripetal force is to be reduced to a fraction whose denominator is A3, and
the numerators are to be made analogous by collating together the homo
logous terms. This will be made plainer by Examples.

EXAMPLE 1. Let us suppose the centripetal force to be uniform,

A3
and therefore as —3 or, writing T — X for A in the numerator, as

T3 — 3TTX + 3TXX — X3

— — — =-. Ihen collating together the correspon-

A3

dent terms of the numerators, that is, those that consist of given quantities,
with those of given quantities, and 'those of quantities not given with those
of quantities not given, it will become RGG — RFF -f- TFF to T3 as —
FFX to 3TTX -f 3TXX — X3, or as — FF to — 3TT + 3TX — XX.
Now since the orbit is supposed extremely near to a circle, let it coincide
with a circle ; and because in that case R and T become equal, and X is
infinitely diminished, the last ratios will be, as RGG to T2, so — FF to —
3TT, or as GG to TT, so FF to 3TT; and again, as GG to FF, so TT
to 3TT, that is, as 1 to 3 ; and therefore G is to F, that is, the angle VC/?
to the angle VCP, as 1 to v/3. Therefore since the body, in an immovable

178 THE MATHEMATICAL PRINCIPLES [BOOK I

ellipsis, in descending from the upper to the lower apsis, describes an angle,
if I may so speak, of ISO deg., the other body in a movable ellipsis, and there
fore in the immovable orbit we are treating of, will in its descent from

180

the upper to the lower apsis, describe an angle VCjt? of — ^ deg. And this

\/o

comes to pass by reason of the likeness of this orbit which a body acted
upon by an uniform centripetal force describes, and of that orbit which a
body performing its circuits in a revolving ellipsis will describe in a quies
cent plane. By this collation of the terms, these orbits are made similar ;
not universally, indeed, but then only when they approach very near to a
circular figure. A body, therefore revolving with an uniform centripetal

180

force in an orbit nearly circular, will always describe an angle of — » deg/, or

v/o

103 deg., 55 m., 23 sec., at the centre; moving from the upper apsis to the
lower apsis when it has once described that angle, and thence returning to
the upper apsis when it has described that angle again ; and so on in in-
finitwn.

EXAM. 2. Suppose the centripetal force to be as any power of the alti-

An

tude A, as, for example, An — 3, or-r^ ; where n — 3 and n signify any in-

A.

dices of powers whatever, whether integers or fractions, rational or surd,
affirmative or negative. That numerator An or T — X|n being reduced to
an indeterminate series by my method of converging series, will become

Tn — >/XTn— T + • — ^ XXTn— 2, &c. And conferring these terms
with the terms of the other numerator RGG — RFF + TFF — FFX, it
becomes as RGG — RFF 4- TFF to Tn, so — FF to — ?/.Tn— r + ?~^

XTn — 2, &c. And taking the last ratios where the orbits approach to
circles, it becomes as RGG to T'1, so — FF to — nT-1— T, or as GG to
T"— ', so FF to ?*Tn— ; and again, GG to FF, so Tn— l to nT"—1, that
is, as 1 to n ; and therefore G is to F, that is the angle VCp to the angle
VCP, as 1 to ^/n. Therefore since the angle VCP, described in the de
scent of the body from the upper apsis to the lower apsis in an ellipsis, is
of 180 deg., the angle VC/?, described in the descent of the body from the
upper apsis to the lower apsis in an orbit nearly circular which a body de
scribes with a centripetal force proportional to the power An — 3, will be equal

ISO

to an angle of - — deg., and this angle being repeated, the body will re-
\/ti

turn from the lower to the upper apsis, and so on in infinitum. As if the
centripetal force be as the distance of the body from the centre, that is, as A,

A4
or -p, n will be equal to 4, and ^/n equal to 2 ; and thereLre the angle

IX.] OF NATURAL PHILOSOPHY. IT9

ISO

between the upper and the lower apsis will be equal to — deg., or 90 deg.

Therefore the body having performed a fourth part of one revolution, will
arrive at the lower apsis, and having performed another fourth part, will
arrive at the upper apsis, and so on by turns in infiuitum. This appears
also from Prop. X. For a body acted on by this centripetal force will re
volve in an immovable ellipsis, whose centre is the centre of force. If the

1 A2

centripetal force is reciprocally as the distance, that is, directly as — or —

£ ' A A"'

ji will be equal to 2 ; and therefore the angle between the upper and lower

180

apsis will be — - deg., or 127 deg., 16 min., 45 sec. ; and therefore a body re
v/2

volving with such a force, will by a perpetual repetition of this angle, move
alternately from the upper to the lower and from the lower to the upper
apsis for ever. So. also, if the centripetal force be reciprocally as the
biquadrate root of the eleventh power of the altitude, that is, reciprocally

as A —, and, therefore, directly as -r-fp or as Ts> n wil* ^e etlual f° \> an(1
4 A^- A

1 Of)

- deg. will be equal to 360 deg. ; and therefore the body parting from
v/ n

the upper apsis, and from thence perpetually descending, will arrive at the
lower apsis when it has completed one entire revolution ; and thence as
cending perpetually, when it has completed another entire revolution, it
will arrive again at the upper apsis ; and so alternately for ever.

EXAM. 3. Taking m and n for any indices of the powers of the alti
tude, and b and c for any given numbers, suppose the centripetal force

6Ara + cA" b into T — X> -f- c into T — X

to be as r^ that is, as

A3 A3

or (by the method of converging series above-mentioned) as

bTm + cTn — m6XT" -1 //cXTn— ' mm — m vvrpm un — n

~~2 -- 0A.A1 — ^—

—

t-XXT" — 2, «fcc.

— T$~— ~ and comparing the terms of the numerators, there will

arise RGG — II FF -f TFF to ^Tm + cT" as — FF to — mbTm — i —

" - « + 2" m bXT" - * + "^p cXTn - » .fee. And tak-

ing the last ratios that arise when the orbits come to a circular form, there
will come forth GG to 6Tm — l -f cTn — 1 as FF to mbTm — l + ncT" — J ;
and again, GG to FF as 6Tm — ' + cTn — ' to mbTn — 1 -f ncTn — \
This proportion, by expressing the greatest altitude CV or T arithmeti
cally by unity, becomes, GG to FF as b -{- c to mb -\- ?/c, and therefore as I

(80 THE MATHEMATICAL PRINCIPLES [BOOK 1

tub ~h nc
to - y—7— — • Whence G becomes to P, that is, the angle VCjo to the an-

f) ~T~ C

gle VCP. as 1 to >/- . -'-. And therefore since the angle VCP between

the upper and the lower apsis, in an immovable ellipsis, is of 180 deg., thr
angle VC/? between the same apsides in an orbit which a body describes

b A m I c A n

with a centripetal force, that is. as - — r§— , will be equal to an angle of

A.

ISO v/— 1~TT~ ; deg. And °y tne same reasoning, if the centripetal force
be as - —73 , the angle between the apsides will be found equal to

fi f*

18o V'' — - — - deg. After the same manner the Problem is solved in
nib — >ic

more difficult cases. The quantity to which the centripetal force is pro
portional must always be resolved into a converging series whose denomi
nator is A*. Then the given part of the numerator arising from that
operation is to be supposed in the same ratio to that part of it which is not
given, as the given part of this numerator RGG — RFF -f TFF — FFX.
is to that part of the same numerator which is not given. And taking
away the superfluous quantities, and writing unity for T, the proportion
of G to F is obtained.

COR. 1 . Hence if the centripetal force be as any power of the altitude,
that power may be found from the motion of the apsides ; and so contra
riwise. That is, if the whole angular motion, with which the body returns
to the same apsis, be to the angular motion of one revolution, or 360 deg.,
MS any number as m to another as n, and the altitude called A ; the force

nn

will be as the power A HSii — 3 of the altitude A; the index of which power is
- — 3. This appears by the second example. Hence it is plain that

the force in its recess from the centre cannot decrease in a greater than a
triplicate ratio of the altitude. A body revolving with such a force, and
parting from the apsis, if it once begins to descend, can never arrive at the
lower apsis or least altitude, but will descend to the centre, describing the
curve line treated of in Cor. 3, Prop. XLL But if it should, at its part
ing from the lower apsis, begin to ascend never so little, it will ascend in
irtfimtifm, and never come to the upper apsis ; but will describe the curve
line spoken of in the same Cor., and Cor. 6, Prop. XLIV. So that where
the force in its recess from the centre decreases in a greater than a tripli
cate ratio of the altitude, the body at its parting from the apsis, will either
descend to the centre, or ascend in iiiftnitum, according as it descends or
Ascends at the beginning of its motion. But if the force in its recess from

"SEC. IX.J OF NATURAL PHILOSOPHY. ISi

the centre either decreases in a less than a triplicate ratio of the altitude,
or increases in any ratio of the altitude whatsoever, the body will never
descend to the centre, but will at some time arrive at the lower apsis ; and,
on the contrary, if the body alternately ascending and descending from one
apsis to another never comes to the centre, then either the force increases
in the recess from the centre, or it decreases in a less than a triplicate ratio
of the altitude; and the sooner the body returns from one apsis to another,
the farther is the ratio of the forces from the triplicate ratio. As if the
body should return to and from the upper apsis by an alternate descent and
ascent in 8 revolutions, or in 4, or 2, or \\ • that is, if m should be to n as 8,

or 4, or 2, or H to 1. and therefore --- 3, be g\ — 3,or TV~ 3, or i — 3, or

mm

3

I - 3 ; then the force will be as A~ ? or AT° "~ 3j or A*~~ 3j or A"" G '

that is. it will be reciprocally as A3 C4' or A3 T^' or A3 4' or A3 ""
If the body after each revolution returns to the same apsis, and the apsis

nn _

remains unmoved, then m will be to n as 1 to 1, and therefore A»"«

will be equal to A 2, or — - ; and therefore the decrease of the forces will

A A

be in a duplicate ratio of the altitude ; as was demonstrated above. If the
body in three fourth parts, or two thirds, or one third, or one fourth part
of an entire revolution, return to the same apsis ; m will be to n as £ or ?

n n i_6 _ 3 9 _ 3 o

or ^ or i to 1, and therefore Amm 3 is equal to A 9 'or A4 or A

_ 3 1 6 _ 3 l_l

' ' or A ; and therefore the force is either reciprocally as A fl or

3 613

A4' or directly as A or A . Lastly if the body in its progress from the
upper apsis to the same upper apsis again, goes over one entire revolution
and three deg. more, and therefore that apsis in each revolution of the body
moves three deg. in consequentia ; then m will be to u as 363 deg. to

360 deg. or as 121 to 120, and therefore Amm will be equal to

2 9_ 5_ 2_ JJ

A "' and therefore the centripetal force will be reciprocally as

^T4"6TT> or reciprocally as A2^4^ very nearly. Therefore the centripetal
force decreases in a ratio something greater than the duplicate ; but ap
proaching 59f times nearer to the duplicate than the triplicate.

COR. 2. Hence also if a body, urged by a centripetal force which is re
ciprocally as the square of the altitude, revolves in an ellipsis whose focus
is in the centre of the forces ; and a new and foreign force should be added
to or subducted from this centripetal force, the motion of the apsides arising
from that foreign force may (by the third Example) be known ; and so on
the contrary. As if the force with which the body revolves in the ellipsis

182 THE MATHEMATICAL PRINCIPLES [BOOK I

oe as -r-r- ; and the foreign force subducted as cA, and therefore the remain-

A .A.

^ c^4

ing force as — -^ ; then (by the third Example) b will be equal to 1.

m equal to 1, and n equal to 4 ; and therefore the angle of revolution be

1 — c

•tween the apsides is equal to 180 <*/- — deg. Suppose that foreign force

to be 357.45 parts less than the other force with which the body revolves
in the ellipsis : that is, c to be -3 }y£ j ; A or T being equal to 1 ; and then

l8(Vl~4c will be 18<Vfff Jf or 180.7623, that is, 180 deg., 45 min.,

44 sec. Therefore the body, parting from the upper apsis, will arrive at
the lower apsis with an angular motion of 180 deg., 45 min., 44 sec , and
this angular motion being repeated, will return to the upper apsis ; and
therefore the upper apsis in each revolution will go forward 1 deg., 31 min.,
28 sec. The apsis of the moon is about twice as swift

So much for the motion of bodies in orbits whose planes pass through
the centre of force. It now remains to determine those motions in eccen
trical planes. For those authors who treat of the motion of heavy bodies
used to consider the ascent and descent of such bodies, not only in a per
pendicular direction, but at all degrees of obliquity upon any given planes ;
and for the same reason we are to consider in this place the motions of
bodies tending to centres by means of any forces whatsoever, when those
bodies move in eccentrical planes. These planes are supposed to be
perfectly smooth and polished, so as not to retard the motion of the bodies
in the least. Moreover, in these demonstrations, instead of the planes upon
which those bodies roll or slide, and which are therefore tangent planes to
the bodies, I shall use planes parallel to them, in which the centres of the
bodies move, and by that motion describe orbits. And by the same method
I afterwards determine the motions of bodies performer1 in curve superficies.

SECTION X.

Of the motion of bodies in given superficies, and of the reciprocal motion
of fnnependulous bodies.

PROPOSITION XL VI. PROBLEM XXXII.

Any kind of centripetal force being supposed, and the centre of 'force •, atfft
any plane whatsoever in which the body revolves, being given, and tint
quadratures of curvilinear figures being allowed; it is required to de
termine the motion of a body going off from a given place., with a
given velocity, in the direction of a given right line in, that plane.

SEC. X.J OF NATURAL PHILOSOPHY- 183

Let S be the centre of force, SC the
least distance of that centre from the given
plane, P a body issuing from the place P
in the direction of the right line PZ, Q,
the same body revolving in its trajectory,
and PQ,R the trajectory itself which is
required to be found, described in that
given plane. Join CQ, Q.S, and if in Q,S
we take SV proportional to the centripetal
force with which the body is attracted to
wards the centre S, and draw VT parallel
to CQ, and meeting SC in T ; then will the force SV be resolved into
two (by Cor. 2, of the Laws of Motion), the force ST, and the force TV ; of
which ST aMracting the body in the direction of a line perpendicular to
that plane, does not at all change its motion in that plane. But the action
c f the other force TV, coinciding with the position of the plane itself, at
tracts the body directly towards the given point C in that plane ; a«d
t icreftre causes the body to move in this plane in the same manner as if
the force S F were taken away, and the body were to revolve in free space
about the centre C by means of the force TV alone. But there being given
the centripetal force TV with which the body Q, revolves in free space
about the given centre C, there is given (by Prop. XLII) the trajectory
PQ.R which the body describes ; the place Q,, in which the body will be
found at any given time ; and, lastly, the velocity of the body in that place
Q,. And so e contra. Q..E.I.

PROPOSITION XLV1L THEOREM XV.

Supposing the centripetal force to be proportional to t/ie distance of the
body from the centre ; all bodies revolving in any planes whatsoever
will describe ellipses, and complete their revolutions in equal times ;
and those which move in right lines, running backwards and forwards
alternately ', will complete ttieir several periods of going and returning
in the same times.

For letting all things stand as in the foregoing Proposition, the force
SV, with which the body Q, revolving in any plane PQ,R is attracted to
wards the centre S, is as the distance SO. ; and therefore because SV and
SQ,, TV and CQ, are proportional, the force TV with which the body is
attracted towards the given point C in the plane of the orbit is as the dis
tance CQ,. Therefore the forces with which bodies found in the plane
PQ,R are attracted towaitis the point O, are in proportion to the distances
equal to the forces with which the same bodies are attract-ed every way to
wards the centre S ; and therefore the bodies will move in the same times,
and in the same figures, in any plane PQR about the point C. n* they

THE MATHEMATICAL PRINCIPLES [BOOK I.

would do in free spaces about the centre S ; and therefore (by Cor. 2, Prop.
X, ai d Cor. 2, Prop. XXXVIII.) they will in equal times either describe
ellipses m that plane about the centre C, or move to and fro in right lines
passing through the centre C in that plane; completing the same periods
of time in all cases. Q.E.D.

SCHOLIUM.

The ascent and descent of bodies in curve superficies has a near relation
to these motions we have been speaking of. Imagine curve lines to be de
scribed on any plane, and to revolve about any given axes passing through
the centre of force, and by that revolution to describe curve superficies ; and
that the bodies move in such sort that their centres may be always found
m those superficies. If those bodies reciprocate to and fro with an oblique
ascent and descent, their motions will be performed in planes passing through
tiie axis, and therefore in the curve lines, by whose revolution those curve
superficies were generated. In those cases, therefore, it will be sufficient to
consider thp motion in those curve lines.

PROPOSITION XL VIII. THEOREM XVI.

If a wheel stands npon the outside of a globe at right angles thereto, and
revolving about its own axis goes forward in a great circle, the length
of lite curvilinear path which any point, given in the perimeter of the
wheel, hath described, since, the time that it touched the globe (which
curvilinear path w~e may call the cycloid, or epicycloid), will be to double
the versed sine of half the arc which since that time has touched the
globe in passing over it, as the sn,m of the diameters of the globe and
the wheel to the semi-diameter of the globe.

PROPOSITION XLIX. THEOREM XVII.

ff a wheel stand upon the inside of a concave globe at right angles there
to, and revolving about its own axis go forward in one of the great
circles of the globe, the length of the curvilinear path which any point,
given in the perimeter of the wheel^ hath described since it toncJied the
globe, imll be to the double of the versed sine of half the arc which in
all that time has touched the globe in passing over it, as the difference
of the diameters of the globe and the wheel to the semi-diameter of the
globe.

Let ABL be the globe. C its centre, BPV the wheel insisting thereon,
E the centre of the wheel, B the point of contact, and P the given point
in the perimeter of the wheel. Imagine this wheel to proceed in the great
circle ABL from A through B towards L, and in its progress to revolve in
such a manner that the arcs AB, PB may be always equal one to the other,
:if;d the given point P in the peri meter of the wheel may describe in thf

SEC. X.I

OF NATURAL PHILOSOPHY.

s

185

H

mean time the curvilinear path AP. Let AP be the whole curvilinear
path described since the wheel touched the globe in A, and the length cf
this path AP will be to twice the versed sine of the arc |PB as 20 E to
CB. For let the right line CE (produced if need be) meet the wheel in V,
and join CP, BP, EP, VP ; produce CP, and let fall thereon the perpen
dicular VF. Let PH, VH, meeting in H, touch the circle in P and V,
and let PH cut YF in G, and to VP let fall the perpendiculars GI, HK.
From the centre C with any interval let there be described the circle wow,
cutting the right line CP in nt the perimeter of the wheel BP in o, and
the curvilinear path AP in m ; and from the centre V with the interval
Vo let there be described a circle cutting VP produced in q.

Because the wheel in its progress always revolves about the point of con
tact B. it is manifest that the right line BP is perpendicular to that curve line
AP which the point P of the wheel describes, and therefore that the right
line VP will touch this curve in the point P. Let the radius of the circle nmn
be gradually increased or diminished so that at last it become equal to the
distance CP ; and by reason of the similitude of the evanescent figure
Pnn-mq, and the figure PFGVI, the ultimate ratio of the evanescent lined ae
Pra, P//, Po, P<y, that is, the ratio of the momentary mutations of the curve
AP, the right line CP, the circular arc BP, and the right line VP, will ••<

iSS THE MATHEMATICAL PRINCIPLES [BOOK 1.

the same as of the lines PV, PF, PG, PI, respectively. But since VF is
perpendicular to OF, and VH to CV, and therefore the angles HVG, VCF
equal: and the angle VHG (because the angles of the quadrilateral figure
HVEP are right in V and P) is equal to the angle CEP, the triangles
V HG, CEP will be similar ; and thence it will come to pass that as EP is
to CE so is HG to HV or HP, and so KI to KP, and by composition or
division as CB to CE so is PI to PK, and doubling the consequents asCB
to 2CE so PI to PV, and so is Pq to Pm. Therefore the decrement of the
line VP, that is, the increment of the line BY— VP to the increment of the
curve line AP is in a given ratio of CB to 2CE, and therefore (by Cor.
Lena. IV) the lengths BY— YP and AP, generated by those increments, arc
in the same ratio. But if BY be radius, YP is the cosine of the angle BYP
or -*BEP, and therefore BY— YP is the versed sine of the same angle, and
therefore in this wheel, whose radius is ^BV, BY— YP will be double the
versed sine of the arc ^BP. Therefore AP is to double the versed sine oi
the arc ^BP as 2CE to CB. Q.E.D.

The line AP in the former of these Propositions we shall name the cy
cloid without the globe, the other in the latter Proposition the cycloid within
the globe, for distinction sake.

COR. 1. Hence if there be described the entire cycloid ASL, and the
same be bisected in S, the lencth of the part PS will be to the length PV
(which is the double of the sine of the angle YBP, when EB is radius) as
2CE to CB, and therefore in a given ratio.

COR. 2. And the length of the semi-perimeter of the cycloid AS will be
equal to a right line which is to the dumeter of the wheel BY as 2CF
toCB.

PROPOSITION L. PROBLEM XXXIII.

To cause a pendulous body to oscillate in a given cycloid.

Let there be given within the globe QYS de-
scribed with the centre C, the cycloid QRS, bi
sected in R, and meeting the superficies of the
globe with its extreme points Q and S on either
hand. Let there be drawn CR birxcting the arc
QS in O, and let it be produced to A in such
sort that CA may be to CO as CO to CR.
About the centre C, with the interval CA, let
there be described an exterior globe DAF ; and
within this globe, by a wheel whose diameter is
AO, let there be described two semi-cycloids AQ,,
AS, touching the interior globe in Q, and S, and meeting the exterior globe
in A. From that point A, with a thread APT in length equal to the line
AR, let the body T depend, and oscillate in such manner between the two

SlCC. X.J OF NATURAL PHILOSOPHY. 187

semi-cycloids AQ, AS, that, as often as the pendulum parts from the per
pendicular AR, the upper part of the thread AP may be applied to that
semi-cycloid APS towards which the motion tends, and fold itself round
that curve line, as if it were some solid obstacle, the remaining part of the
same thread PT which has not yet touched the semi-cycloid continuing
straight. Then will the weight T oscillate in the given cycloid QRS.

Q.E.F.

For let the thread PT meet the cycloid QRS in T, and the circle QOS
m V, and let 0V be drawn j and to the rectilinear part of the thread PT
from the extreme points P and T let there be erected the perpendiculars
BP, T W, meeting the right line CV in B and W. It is evident, from the
construction and generation of the similar figures AS, SR, that those per
pendiculars PB, TVV, cut off from CV the lengths VB, VVV equal the
diameters of the wheels OA, OR. Therefore TP is to VP (which is dou
ble the sine of the angle VBP when ^BV is radius) as BYV to BV, or AO
-f-OR to AO, that is (since CA and CO, CO and CR; and by division AO
and OR are proportional), as CA + CO to CA, or, if BV be bisected in E,
as 2CE to CB. Therefore (by Cor. 1, Prop. XLIX), the length of the
rectilinear part of the thread PT is always equal to the arc of the cycloid
PS, and the whole thread APT is always equal to the half of the cycloid
APS, that is (by Cor. 2, Prop. XLIX), to the length AR. And there
fore contrariwise, if the string remain always equal to the length AR, the
point T will always move in the given cycloid QRS. Q.E.D.

COR. The string AR is equal to the semi-cycloid AS, and therefore has
the same ratio to AC the semi-diameter of the exterior globe as the like
semi-cycloid SR has to CO the semi-diameter of the interior globe.

PROPOSITION LI. THEOREM XVIII.

If a centripetal force tending on all sides to the centre C of a globe, be in
all places as the distance of the place from the centre, and by this force
alone acting upon it, the body T oscillate (in the manner above de
scribed] in the perimeter of the cycloid QRS ; / say, that all the oscil
lations, how unequal soever in tfiemselves, will be performed in equal
times.

For upon the tangent T W infinitely produced let fall the perpendicular
CX, and join CT. Because the centripetal force with which the body T
is impelled towards C is as the distance CT, let this (by Cor. 2, of the
I ,aws) be resolved into the parts CX, TX, of which CX impelling the
body directly from P stretches the thread PT, and by the resistance the
'rhread makes to it is totally employed, producing no other effect ; but the
3ther part TX, impelling the body transversely or towards X, directly
accelerates the motion in the cycloid. Then it is plain that the accelera
tion of the body, proportional to this accelerating force, will bo every

188

THE MATHEMATICAL PRINCIPLES

[BOOK 1

moment as the length TX, that is (because CV\
WV, and TX, TW proportional to them are given),
as the length TW, that is (by Cor. 1, Prop. XLIX)
as the length of the arc of the cycloid TR. If there
fore two pendulums APT, Apt, be unequally drawn
aside from the perpendicular AR, and let fall together,
their accelerations will be always as the arcs to be de
scribed TR, tR. But the parts described at the
beginning of the motion are as the accelerations, thai
is, as the wholes that are to be described at the be
ginning, and therefore the parts which remain to be
described, and the subsequent accelerations proportional to those parts, are
also as the wholes, and so on. Therefore the accelerations, and consequently
the velocities generated, and the parts described with those velocities, and
the parts to be described, are always as the wholes ; and therefore the parts
to be described preserving a given ratio to each other will vanish together,
that is, the two bodies oscillating will arrive together at the perpendicular AR.
And since on the other hand the ascent of the pendulums from the lowest place
R through the same cycloidal arcs with a retrograde motion, is retarded in
the several places they pass through by the same forces by which their de
scent was accelerated : it is plain that the velocities of their ascent and de
scent through the same arcs are equal, and consequently performed in equal
times ; and, therefore, since the two parts of the cycloid RS and RQ lying
on either side of the perpendicular are similar and equal, the two pendu
lums will perform as well the wholes as the halves of their oscillations in
the same times. Q.E.D.

' COR. The force with which the body T is accelerated or retarded in any
place T of the cycloid, is to the whole weight of the same body in the
highest place S or Q, as the arc of the cycloid TR is to the arc SR or QR

PROPOSITION LIL PROBLEM XXXIV.

To define the velocities of the pendulums in the several places, and the
times in which both the entire oscillations, and the several parts of them
are performed.

About any centre G, with the interval GH equal to
the arc of the cycloid RS, describe a semi-circle HKM
bisected by the semi-diameter GK. And if a centripe
tal force proportional to the distance of the places from
the centre tend to the centre G, and it be in the peri
meter HIK equal to the centripetal force in the perime
ter of the globe Q,OS tending towards its centre, and at
the same time that the pendulum T is let fall from the
highest place S, a body, as L, is let fall from H to G ; then because th<

SEC. X.J OF NATURAL PHILOSOPHY. 189

forces which act upon the bodies are equal at the be
ginning, and always proportional to the spaces to be
described TR, LG, and therefore if TR and LG are
equal, are also equal in the places T and L, it is plain
that those bodies describe at the beginning equal spaces M
ST, HL, and therefore are still acted upon equally, and continue to describe
equal spaces. Therefore by Prop. XXXVIII, the time in which the body
describes the arc ST is to the time of one oscillation, as the arc HI the time
in which the body H arrives at L, to the semi-periphery HKM, the time
in which the body H will come to M. And the velocity of the pendulous
body in the place T is to its velocity in the lowest place R, that is, the
velocity of the body H in the place L to its velocity in the place G, or the
momentary increment of the line HL to the momentary increment of the
line HG (the arcs HI, HK increasing with an equable flux) as the ordinato
LI to the radius GK. or as v/SR2 — Til2 to SR. Hence, since in unequal
oscillations there are described in equal time arcs proportional to the en
tire arcs of the oscillations, there are obtained from the times given, both
the velocities and the arcs described in all the oscillations universally.
Which was first required.

Let now any pendulous bodies oscillate in different cycloids described
within different globes, whose absolute forces are also different ; and if the
absolute force of any globe Q.OS be called V, the accelerative force with
which the pendulum is acted on in the circumference of this globe, when it
begins to move directly towards its centre, will be as the distance of the
pendulous body from that centre and the absolute force of the globe con-
junctly, that is, as CO X V. Therefore the lineola HY, which is as this
accelerated force CO X V, will be described in a given time : and if there
be erected the perpendicular YZ meeting the circumference in Z, the nascent
arc HZ will denote that given time. But that nascent arc HZ is in the
subduplicate ratio of the rectangle GHY, and therefore as v/GH X CO X V
Whence the time of an entire oscillation in the cycloid Q,RS (it being as
the semi-periphery HKM, wrhich denotes that entire oscillation, directly :
and as the arc HZ which in like manner denotes a given time inversely)
will be as GH directly and v/GH X CO X V inversely ; that is, because

GH and SR are equal, as Vnr, . ™ or (by Cor. Prop. L,) as X/-TTVT- —

UU X V AO X V

Therefore the oscillations in all globes and cycloids, performed with what
absolute forces soever, are in a ratio compounded of the subduplicate ratio of
the length of the string directly, and the subduplicate ratio of the distance
between the point of suspension and the centre of the globe inversely, and
the subduplicate ratio of the absolute force of the globe inversely also
Q.E.I.

t90 THE MATHEMATICAL PRINCIPLES [Bo^K 1.

COR. 1. Hence also the times of oscillating, falling, and revolving bodies
may be compared among themselves. For if the diameter of the wheel
with which the cycloid is described within the globe is supposed equal to
the semi-diameter of the globe, the cycloid will become a right line passing
through the centre of the globe, and the oscillation will be changed into a
descent and subsequent ascent in that right line. Whence there is given
both the time of the descent from any place to the centre, and the time equal
to it in which the body revolving uniformly about the centre of the globe
at any distance describes an arc of a quadrant For this time (by
Case 2) is to the time of half the oscillation in any cycloid QJR.S as 1 to

AR
V AC'

COR. 2. Hence also follow what Sir Christopher Wren and M. Huygevs
have discovered concerning the vulgar cycloid. For if the diameter of the
globe be infinitely increased, its sphacrical superficies will be changed into a
plane, and the centripetal force will act uniformly in the direction of lines
perpendicular to that plane, and this cycloid of our's will become the same
with the common cycloid. But in that case the length of the arc of the
cycloid between that plane and the describing point will become equal to
four times the versed sine of half the arc of the wheel between the same
plane and the describing point, as was discovered by Sir Christopher Wren.
And a pendulum between two such cycloids will oscillate in a similar and
equal cycloid in equal times, as M. Huygens demonstrated. The descent
of heavy bodies also in the time of one oscillation will be the same as M.
Huygens exhibited.

The propositions here demonstrated are adapted to the true constitution
of the Earth, in so far as wheels moving in any of its great circles will de
scribe, by the motions of nails fixed in their perimeters, cycloids without the
globe ; and pendulums, in mines and deep caverns of the Earth, must oscil
late in cycloids within the globe, that those oscillations may be performed
in equal times. For gravity (as will be shewn in the third book) decreases
in its progress from the superficies of the Earth ; upwards in a duplicate
ratio of the distances from the centre of the Earth ; downwards in a sim
ple ratio of the same.

PROPOSITION LIII. PROBLEM XXXV.

Granting the quadratures of curvilinear figures, it is required to find
the forces with which bodies moving in given curve lines may always
perform their oscillations in equal times.

Let the body T oscillate in any curve line STRQ,, whose axis is AR
passing through the centre of force C. Draw TX touching that curve in
any place of the body T, and in that tangent TX take TY equal to the
arc TR. The length of that arc is known from the common methods used

SEC. X.

OF NATURAL PHILOSOPHY.

191

for the quadratures of figures. From the point Y
draw the right line YZ perpendicular to the tangent.
Draw CT meeting that perpendicular in Z, and the
centripetal force will be proportional to the right line
TZ. Q.E.I.

For if the force with which the body is attracted
from T towards C be expressed by the right line TZ
taken proportional to it, that force will be resolved
into two forces TY, YZ, of which YZ drawing the
body in the direction of the length of the thread PT,
docs not at all change its motion ; whereas the other
force TY directly accelerates or retards its mction in the curve STRQ.
Wherefore since that force is as the space to be described TR, the acceler
ations or retardations of the body in describing two proportional parts (u
greater arid a less) of two oscillations, will be always as those parts, and
therefore will cause those parts to be described together. But bodies which
continually describe together parts proportional to the wholes, will describe
the wholes together also. Q,.E.l).

COR. 1. Hence if the body T, hanging by a rectilinear thread
AT from the centre A, describe the circular arc STRQ,,
and in the mean time be acted on by any force tending
downwards with parallel directions, which is to the uni
form force of gravity as the arc TR to its sine TN, the
times of the several oscillations will be equal. For because
TZ, AR are parallel, the triangles ATN, ZTY are similar ; and there
fore TZ will be to AT as TY to TN ; that is, if the uniform force of
gravity be expressed by the given length AT, the force TZ. by which the
oscillations become isochronous, will be to the force of gravity AT, as the
arc TR equal to TY is to TN the sine of that arc.

COR. 2. And therefore in clocks, if forces were impressed by some ma
chine upon the pendulum which preserves the motion, and so compounded
with the force of gravity that the whole force tending downwards should
be always as a line produced by applying the rectangle under the arc TR
and the radius AR to the sine TN, all the oscillations will become
isochronous.

PROPOSITION LIV. PROBLEM XXX YI.

Granting the quadratures of curvilinear figures, it is required to find
the times in which bodies by means of any centripetal force will descend
or ascend in any curve lines described in, a plane passing through the
centre of force.
Let the body descend from any place S, and move in any curve ST/R

given in a plane passing through the centre of force C. Join CS, and lei

192

THE MATHEMATICAL PRINCIPLES

[BOOK 1

Q it be divided into innumerable equal parts, and let
Dd be one of those parts. From the centre C, with
the intervals CD, Cd, let the circles DT, dt be de
scribed, meeting the curve line ST*R in T and t.
And because the law of centripetal force is given.
and also the altitude CS from which the body at
first fell, there will be given the velocity of the body
in any other altitude CT (by Prop. XXXIX). But
the time in which the body describes the lineola Tt
is as the length of that lineola, that is, as the secant
of the angle /TC directly, and the velocity inversely.
Lei, the ordinate DN, proportional to this time, be made perpendicular to
the right line CS at the point D, and because Dd is given, the rectangle
Dd X DN, that is, the area DNwc?, will be proportional to the same time.
Therefore if PN/?, be a curve line in which the point N is perpetually found,
and its asymptote be the right line SQ, standing upon the line CS at right
angles, the area SQPJN D will be proportional to the time in which the body
in its descent hath described the line ST ; and therefore that area being
found, the time is also given. Q.E.I.

PROPOSITION LV. THEOREM XIX.

If a body move in any curve superficies, whose axis passes through the
centre of force, and from the body a perpendicular be let fall iipon the
axis \ and a line parallel and equal thereto be drawn from any given
point of the axis ; I say, that this parallel line will describe an area
proportional to the time.

Let BKL be a curve superficies, T a body
revolving in it, STR a trajectory which the
body describes in the same, S the beginning
of the trajectory, OMK the axis of the curve
superficies, TN a right line let fall perpendic
ularly from the body to the axis ; OP a line
parallel and equal thereto drawn from the
given point O in the axis ; AP the orthogra
phic projection of the trajectory described by
the point P in the plane AOP in which the
revolving line OP is found : A the beginning
of that projection, answering to the point S ;
TO a right line drawn from the body to the centre ; TG a part thereof
proportional to the centripetal force with which the body tends towards the
centre C ; TM a right line perpendicular to the curve superficies ; TI a
part thereof proportional to the force of pressure with which the body urges

SEC. X.]

OF NATURAL PHILOSOPHY.

193

the superficies, and therefore with which it is again repelled by the super
ficies towards M ; PTF a right line parallel to the axis and passing through
the body, and OF, IH right lines let fall perpendicularly from the points
G and I upon that parallel PHTF. I say, now. that the area AGP, de
scribed by the radius OP from the beginning of the motion, is proportional
to the time. For the force TG (by Cor. 2, of the Laws of Motion) is re
solved into the forces TF, FG ; and the force TI into the forces TH, HI ;
but the forces TF, TH, acting in the direction of the line PF perpendicular
to the plane AOP, introduce no change in the motion of the body but in a di
rection perpendicular to that plane. Therefore its motion, so far as it has
the same direction with the position of the plane, that is, the motion of the
point P, by which the projection AP of the trajectory is described in that
plane, is the same as if the forces TF, TH were taken away, and the body
wei e acted on by the forces FG, HI alone ; that is, the same as ,f the body
were to describe in the plane AOP the curve AP by means of a centripetal
force tending to the centre O, and equal to the sum of the forces FG and
HI. But with such a force as that (by Prop. 1) the area AOP will be de
scribed proportional to the time. Q.E.D.

COR. By the same reasoning, if a body, acted on by forces tending to
two or more centres in any the same right line CO, should describe in a
free space any curve line ST, the area AOP would be always proportional
to the time.

PROPOSITION LVI. PROBLEM XXXVII.

Granting the quadratures of curvilinear figures, and supposing that
there are given both the law of centripetal force tending to a given cen
tre, and the curve superficies whose axis passes through that centre ;
it is required to find the trajectory which a body will describe in that
superficies, when going off from a given place with a given velocity,
and in a given direction in that superficies.
The last construction remaining, let the
body T go from the given place S, in the di
rection of a line given by position, and turn
into the trajectory sought STR, whose ortho
graphic projection in the plane BDO is AP.
And from the given velocity of the body in
the altitude SC, its velocity in any other al
titude TC will be also given. With that
velocity, in a given moment of time, let the
body describe the particle Tt of its trajectory,
and let P/? be the projection of that particle
described in the plane AOP. Join Op, and
a little circle being described upon the curve superficies about the centre T

13

194 THE MATHEMATICAL PRINCIPLES [BOOK I

with the interval TV let the projection of that little circle in the plane AOP
be the ellipsis pQ. And because the magnitude of that little circle T/, and
TN or PO its distance from the axis CO is also given, the ellipsis pQ, will
be given both in kind and magnitude, as also its position to the right line
PO. And since the area PO/? is proportional to the time, and therefore
given because the time is given, the angle POp will be given. And thence
will be given jo the common intersection of the ellipsis and. the right line
Op, together with the angle OPp, in which the projection APp of the tra
jectory cuts the line OP. But from thence (by conferring Prop. XLI, with
Us 2d Cor.) the mariner of determining the curve APp easily appears.
Then from the several points P of that projection erecting to the plane
AOP, the perpendiculars PT meeting the curve superficies in T, there will
be £iven the several points T of the trajectory. Q.E.I.

SECTION XL

f 'f the motions of bodies tending' to each other with centripetal forces.
I have hitherto been treating of the attractions of bodies towards an im
movable centre; though very probably there is no such thing existent in
nature. For attractions are made towards bodies, and the actions of the
bodies attracted and attracting are always reciprocal and equal, by Law III ;
BO that if there are two bodies, neither the attracted nor the attracting body
is truly at rest, but both (by Cor. 4, of the Laws of Motion), being as it
were mutually attracted, revolve about a common centre of gravity. And
if there be more bodies, which are either attracted by one single one which
is attracted by them again, or which all of them, attract each other mutu
ally , these bodies will be so moved among themselves, as that their common
centre of gravity will either be at rest, or move uniformly forward in a
right line. I shall therefore at present go on to treat of the motion of
bodies mutually attracting each other ; considering the centripetal forces
as attractions ; though perhaps in a physical strictness they may more truly
be called impulses. But these propositions are to be considered as purely
mathematical; and therefore, laying aside all physical considerations, I
make use of a familiar way of speaking, to make myself the more easily
understood by a mathematical reader.

PROPOSITION LVII. THEOREM XX.

Two bodies attracting each other mutually describe similar figures about

their common centre of gravity, and about each other mutually.

For the distances of the bodies from their common centre of gravity are

leciprocally as the bodies; and therefore in a given ratio to each other:

*nd thence, bv composition of ratios, in a given ratio to the whole distance

SEC. XL] OF NATURAL PHILOSOPHY. 195

between the bodies. Now these distances revolve about their common term
with an equable angular motion, because lying in the same right line they
never change their inclination to each other mutually But right lines
that are in a given ratio to each other, and revolve about their terms with
an equal angular motion, describe upon planes, which either rest with
those terms, or move with any motion not angular, figures entirely similar
round those terms. Therefore the figures described by the revolution ot
these distances are similar. Q.E.D.

PROPOSITION LVIll. THEOREM XXI.

If two bodies attract each other mutually with forces of any kind, and
in the mean time revolve about the common centre of gravity ; I say,
that, by the same forces, there may be described round either body un
moved ajigure similar and equal to the figures ivhich the bodies so
moving describe round each other mutually.
Let the bodies S and P revolve about their common centre of gravity

C, proceeding from S to T, and from P to Q,. From the given point s lot

there be continually drawn sp, sq, equal and parallel to SP, TQ, ; and the
;urve pqv, which the point p describes in its revolution round the immovable
point s, will be similar and equal to the curves which the bodies S and P'
describe about each other mutually ; and therefore, by Theor. XX, similar
to the curves ST and PQ,V which the same bodies describe about their
common centre of gravity C • and that because the proportions of the lines
SC. CP, and SP or sp, to each other, are given.

CASE 1. The common centre of gravity C (by Cor. 4, of the Laws of Mo
tion) is either at rest, or moves uniformly in a right line. Let us first
suppose it at rest, and in s and p let there be placed two bodies, one im
movable in s, the other movable in p, similar and equal to the bodies S arid
P. Then let the right lines PR and pr touch the curves PQ, and pq ki P
and p, and produce CQ, and sq to R and r. And because the figures
CPRQ, sprq are similar, RQ, will be to rq as CP to sp, and therefore in a
given ratio. Hence if the force with which the body P is attracted to
wards the body S, and by consequence towards the intermediate point the
centre C, were to the force with which the body p is attracted towards the
Centre 5. in the same given ratio, these forces would in equal times attract

196 THE MATHEMATICAL PRINCIPLES |BoOK 1

the bodies from the tangents PR, pr to the arcs PQ, pq, through the in
tervals proportional to them RQ,, rq ; and therefore this last force (tending
to s) would make the body p revolve in the curve pqv, which would becomr
similar to the curve PQV, in which the first force obliges the body P i(
revolve ; and their revolutions would be completed in the same timeg
But because those forces are not to each other in the ratio of CP to sp, bu;
(by reason of the similarity and equality of the bodies S and s, P and /
and the equality of the distances SP, sp) mutually equal, the bodies ii
equal times will be equally drawn from the tangents; and therefore tLV
the body p may be attracted through the greater interval rq, there is re
quired a greater time, which will be in the subduplicate ratio of the inter
vals ; because, by Lemma X, the spaces described at the very beginning ol
the motion are in a duplicate ratio of the times. Suppose, then the velocity
of the body p to be to the velocity of the body P in a subduplicate ratio of
the distance sp to the distance CP, so that the arcs pq, PQ, which are in a
simple proportion to each other, may be described in times that are in n
subduplicate ratio of the distances ; and the bodies P, p, always attracted
by equal forces, will describe round the quiescent centres C and s similar
figures PQV, pqv, the latter of which pqv is similar and equal to the figure
ivhich the body P describes round the movable body S. Q.E.I)

CASE 2. Suppose now that the common centre of gravity, together with
the space in which the bodies are moved among themselves, proceeds uni
formly in a right line ; and (by Cor. 6, of the Laws of Motion) all the mo
tions in this space will be performed in the same manner as before ; and
therefore the bodies will describe mutually about each other the same fig
ures as before, which will be therefore similar and equal to the figure pqv.
Q.E.D.

COR. 1. Hence two bodies attracting each other with forces proportional
to their distance, describe (by Prop. X) both round their common centre ol
gravity, and round each other mutually concentrical ellipses ; and, vice
versa, if such figures are described, the forces are proportional to the dis
tances.

COR. 2. And two bodies, whose forces are reciprocally proportional to
the square of their distance, describe (by Prop. XI, XII, XIII), both round
their common centre of gravity, and round each other mutually, conic sec
tions having their focus in the centre about which the figures are described.
And, vice versa, if such figures are described, the centripetal forces are re
ciprocally proportional to the squares of the distance.

COR. 3. Any two bodies revolving round their common centre of gravity
describe areas proportional to the times, by radii drawn both to that centre
and to each other mutually-

>EC. XL] OF NATURAL PHILOSOPHY. 197

PROPOSITION LIX. THEOREM XXII.

The periodic time of two bodies S and P revolving round their common
centre of gravity C,is to the periodic time of one of the bwlies 1? re
volving round the other S remaining unmoved, and describing a fig
ure similar and equal to those which the bodies describe about each
other mutually r, in a subduplicate ratio of the other body S to the sii/rn
of the bodies S -f P.

For, by the demonstration of the last Proposition, the times in which
any similar arcs PQ and pq are described are in a subduplicate ratio of the
distances CP and SP, or sp, that is, in a subduplicate ratio of the ody S
to the sum of the bodies S + P. And by composition of ratios, the sums
of the times in which all the similar arcs PQ and pq are described, that is,
the whole times in which the whole similar figures are described are in the
same subduplicate ratio. Q.E.D.

PROPOSITION LX. THEOREM XXIII.

If two bodies S and P, attracting each other with forces reciprocally pro
portional to the squares of their distance, revolve about their common
centre of gravity ; I say, that the principal axis of the ellipsis which
either of the bodies, as P, describes by this motion about the other S,
will be to the principal axis of the ellipsis, which the same body P may
describe in the same periodical time about the other body S quiescent,
as the sum of the two bodies S + P to the first of two mean, propor
tionals between that sum and the other body S.

For if the ellipses described were equal to each other, their periodic times
by the last Theorem would be in a subduplicate ratio of the body S to the
sum of the bodies S 4- P. Let the periodic time in the latter ellipsis be
diminished in that ratio, and the periodic times will become equal ; but,
by Prop. XV, the principal axis of the ellipsis will be diminished in a ratio
sesquiplicate to the former ratio ; that is, in a ratio to which the ratio of
S to S 4- P is triplicate ; and therefore that axis will be to the principal
axis of the other ellipsis as the first of two mean proportionals between S
-f- P and S to S 4- P. And inversely the principal axis of the ellipsis de
scribed about the movable body will be to the principal axis of that described
round the immovable as S + P to the first of two mean proportionals be
tween S 4- P and S. Q.E.D.

PROPOSITION LXI. THEOREM XXIV.

If two bodies attracting each other with any kind of forces, and not
otherwise agitated or obstructed, are moved in any manner whatsoever,
those 'motions will be the same as if they did not at all attract each
other mutually, but were both attracted with the same forces by a third
body placed in their common centre of gravity ; and the law of the

198 THE MATHEMATICAL PRINCIPLES [BOOK I

attracting Jones will be the sam# in respect of the distance of the.

bodies from, the common centre, as in respect of the distance between

the two bodies.

For those forces with which the bodies attract each other mutually, by
tending to the bodies, tend also to the common centre of gravity lying di
rectly between them ; and therefore are the same as if they proceeded from
'an intermediate body. QJG.D.

And because there is given the ratio of the distance of either body from
that common centre to the distance between the two bodies, there is given,
of course, the ratio of any power of one distance to the same power of the
. ther distance ; and also the ratio of any quantity derived in any manner
from one of the distances compounded any how with given quantities, to
another quantity derived in like manner from the other distance, and as
many given quantities having that given ratio of the distances to the first
Therefore if the force with which one body is attracted by another be di
rectly or inversely as the distance of the bodies from each other, or as any
power of that distance ; or, lastly, as any quantity derived after any man
ner from that distance compounded with given q-uantities ; then will the
same force with which the same body is attracted to the common centre of
gravity be in like manner directly or inversely as the distance of the at
tracted body from the common centre, or as any power of that distance ;
or, lastly, as a quantity derived in like sort from that distance compounded
with analogous given quantities. That is, the law of attracting force will
be the same with respect to both distances. Q,.E.D.

PROPOSITION LXII. PROBLEM XXXVIII.

To determine the motions of two bodies which attract each other with
forces reciprocally proportional to the squares of the distance between
them, aflid are, let fall from given places.

The bodies, by the last Theorem, will be moved in the same manner as
if they were attracted by a third placed in the common centre of their
gravity ; and by the hypothesis that centre will be quiescent at the begin
ning of their motion, and therefore (by Cor. 4, of the Laws of Motion) will
be always quiescent. The motions of the bodies are therefore to be deter
mined (by Prob. XXV) in the same manner as if they were impelled by
forces tending to that centre: and then we shall have the motions of the
bodies attracting each other mutually. Q.E.I.

PROPOSITION LXIII. PROBLEM XXXIX.

To determine the motions of two bodies attracting each other with forces
reciprocally proportional to the squares of their distance, and going
off from given places in, given directions with given velocities.
The motions of the bodies at the beginning being given, there is given

SEC. XL] OF NATURAL PHILOSOPHY. 1 %

also the uniform motion of the common centre of gravity, and the motion
of the space which moves along with this centre uniformly in a right line,
and also the very first, or beginning motions of the bodies in respect of this
space. Then (by Cor. 5, of the Laws, and the last Theorem) the subse
quent motions will be performed in the same manner in that space, as if
that space together with the common centre of gravity were at rest, and as
if the bodies did not attract each other, but were attracted by a third body
placed in that centre. The motion therefore in this movable space of each
body going off from a given place, in a given direction, with a given velo
city, and acted upon by a centripetal force tending to that centre, is to be
determined by Prob. IX and XXVI, and at the same time will be obtained
the motion of the other round the same centre. With this motion com
pound the uniform progressive motion of the entire system of the space and
the bodies revolving in it, and there will be obtained the absolute motion
of the bodies in immovable space. Q..E.I.

PROPOSITION LXIV. PROBLEM XL.

Supposing forces with which bodies mutually attract each other to in
crease in a simple ratio of their distances from the centres ; it is ro-
quired to find the motions of several bodies among themselves.
Suppose the first two bodies T and L
to have their common centre of gravity in
L). These, by Cor. 1, Theor. XXI, will S
describe ellipses having their centres in D,
the magnitudes of which ellipses are

known by Prob. V. J- -- \- ? L

Let now a third body S attract the two
former T and L with the accelerative forces ST, SL, and let it be attract
ed again by them. The force ST (by Cor. 2, of the Laws of Motion) is
resolved into the forces SD, DT ; and the force SL into the forces SD and
DL. Now the forces DT, DL. which are as their sum TL, and therefore
as the accelerative forces with which the bodies T and L attract each other
mutually, added to the forces of the bodies T and L, the first to the first,
and the last to the last, compose forces proportional to the distances DT
and DL as before, but only greater than those former forces : and there
fore (by Cor. 1, Prop. X, and Cor. l,and 8, Prop. IV) they will cause those
bodies to describe ellipses as before, but with a swifter motion. The re
maining accelerative forces SD and DL, by the motive forces SD X Tand
SD X L, which are as the bodies attracting those bodies equally and in the
direction of the lines TI, LK parallel to DS, do not at all change their situ
ations with respect to one another, but cause them equally to approach to
the line IK ; which must be imagined drawn through the middle of the
body S, and perpendicular to the line DS. But that approach to the line

200 THE MATHEMATICAL PRINCIPLES [BoOK I.

IK will be hindered by causing the system of the bodies T and L on one
side, and the body S on the other, with proper velocities, to revolve round
the common centre of gravity C. With such a motion the body S, because
the sum of the motive forces SD X T and SD X L is proportional to the
distance OS, tends to the centre C, will describe an ellipsis round the same
centre C; and the point D, because the lines CS and CD are proportional,
will describe a like ellipsis over against it. But the bodies T and L, at
tracted by the motive forces SD X T and SD X L, the first by the first,
and the last by the last, equally and in the direction of the parallel lines TI
and LK, as was said before, will (by Cor. 5 and 6, of the Laws of Motion)
continue to describe their ellipses round the movable centre D, as before.
Q.E.I.

Let there be added a fourth body V, and, by the like reasoning, it will
be demonstrated that this body and the point C will describe ellipses about
the common centre of gravity B ; the motions of the bodies T, L, and S
round the centres D and C remaining the same as before ; but accelerated.
Arid by the same method one may add yet more bodies at pleasure. Q..E.I.
^This would be the case, though the bodies T and L attract each other
mutually with accelerative forces either greater or less than those with
which they attract the other bodies in proportion to their distance. Let
all the mutual accelerative attractions be to each other as the distances
multiplied into the attracting bodies ; and from what has gone before it
will easily be concluded that all the bodies will describe different ellipses
with equal periodical times about their common centre of gravity B, in an
immovable plane. Q.E.I.

PROPOSITION LXV. THEOREM XXV.

Bodies, whose forces decrease in a duplicate ratio of their distances from
their centres, 'may move among" themselves in ellipses ; and by radii
drawn to the foci may describe areas proportional to the times very
nearly.

In the last Proposition we demonstrated that case in which the motions
will be performed exactly in ellipses. The more distant the law of the
forces is from the law in that case, the more will the bodies disturb each
other's motions ; neither is it possible that bodies attracting each other
mutually according to the law supposed in this Proposition should move
exactly in ellipses, unless by keepirg a certain proportion of distances from
each other. However, in the following crises the orbits will not much dif
fer from ellipses.

CASE I. Imagine several lesser bodies to revolve about some very great
one at different distances from it, and suppose absolute forces tending to
rvery one of the bodies proportional to each. And because (by Cor. 4, ol
the I aws) the common centre of gravity of them all is either at rest, 01

iSEC. XL] OF NATURAL PHILOSOPHY. 20 i

moves uniformly forward in a right line, suppose the lesser bodies so small
that the groat body may be never at a sensible distance from that centre ;
and then the great body will, without any sensible error, be either at rest,
or move uniformly forward in a right line; and the lesser will revolve
about that great one in ellipses, and by radii drawn thereto will describe
areas proportional to the times ; if we except the errors that may be intro
duced by the receding of the great body from the common centre of gravity,
or by the mutual actions of the lesser bodies upon each other. But the
lesser bodies may be so far diminished, as that this recess and the mutual
actions of the bodies on each other may become less than any assignable;
and therefore so as that the orbits may become ellipses, and the areas an
swer to the times, without any error that is not less than any assignable.
Q.E.O.

CASE 2. Let us imagine a system of lesser bodies revolving about a very
great one in the manner just described, or any other system of two bodies
revolving about each other to be moving uniformly forward in a right line, and
in the mean time to be impelled sideways by the force of another vastly greater
body situate at a great distance. And because the equal accelerative forces
with which the bodies are impelled in parallel directions do not change the
situation of the bodies with respect to each other, but only oblige the whole
system to change its place while the parts still retain their motions among
themselves, it is manifest that no change in those motions of the attracted
bodies can arise from their attractions towards the greater, unless by the
inequality of the accelerative attractions, or by the inclinations of the lines
towards each other, in whose directions the attractions are made. Suppose,
therefore, all the accelerative attractions made towards the great body
to be among themselves as the squares of the distances reciprocally ; and
then, by increasing the distance of the great body till the differences of fhe
right lines drawn from that to the others in respect of their length, and the
inclinations of those lines to each other, be less than any given, the mo
tions of the parts of the system will continue without errors that are not
less than any given. And because, by the small distance of those parts from
each other, the whole system is attracted as if it were but one body, it will
therefore be moved by this attraction as if it were one body ; that is, its
centre of gravity will describe about the great bod/ one of the conic sec
tions (that is, a parabola or hyperbola when the attraction is but languid
and an ellipsis when it is more vigorous) ; and by radii drawn thereto, it
will describe areas proportional to the times, without any errors but thos«
which arise from the distances of the parts, which are by the supposition
exceedingly small, and may be diminished at pleasure. Q,.E.O.

By a like reasoning one may proceed to more compounded cases in in-
finitum.

COR 1. In the second Case, the nearer the very great body approaches to

^0'^ THE MATHEMATICAL PRINCIPLES [CoOK I

the system of two or more revolving bodies, the greater will the pertur
bation be of the motions of the parts of the system among themselves; be
cause the inclinations of the lines drawn from that great body to those
parts become greater ; and the inequality of the proportion is also greater.

COR. 2. But the perturbation will be greatest of all, if we suppose the
uccelerative attractions of the parts of the system towards the greatest body
of all are not to each other reciprocally as the squares of the distances
from that great body ; especially if the inequality of this proportion be
greater than the inequality of the proportion of the distances from the
great body. For if the accelerative force, acting in parallel directions
and equally, causes no perturbation in the motions of the parts of the
system, it must of course, when it acts unequally, cause a perturbation some
where, which will be greater or less as the inequality is greater or less.
The excess of the greater impulses acting upon some bodies, and not acting
upon others, must necessarily change their situation among themselves. And
this perturbation, added to the perturbation arising from the inequality
and inclination of the lines, makes the whole perturbation greater.

COR. '*. Hence if the parts of this system move in ellipses or circles
without any remarkable perturbation, it is manifest that, if they are at all
impelled by accelerative forces tending to any other bodies, the impulse is
very weak, or else is impressed very near equally and in parallel directions
upon all of them.

PROPOSITION LXVL THEOREM XXVI.

Tf three bodies whose forces decrease in a duplicate ratio of the distances
attract each other mutually ; and the accelerative attractions of any
two towards the third be between themselves reciprocally as the squares,
of the distances ; and the two least revolve about the greatest ; I say,
that the interior of the tivo revolving bodies will, by radii drawn to the
innermost and greatest, describe round thai body areas more propor
tional to the times, and a figure more approaching to that of an ellip
sis having its focus in the point of concourse of the radii, if that great
body be agitated by those attractions, than it would do if lhat great
body were not attracted at all by the lesser, but remained at rest ; or
than it would if that great body were very much more or very much
less attracted, <>r very much more or very much less agitated, by the
attractions.
This appears plainly enough from the demonstration of the second

Corollary of tl.e foregoing Proposition; but it may be made out after

this manner by a way of reasoning more distinct and more universally

convincing.

CASE 1. Let the lesser bodies P and S revolve in the same plane about

the greatest body T, the body P describing the interior orbit PAB, and S

SEC. XI.J OF NATURAL PHILOSOPHY. 203

the exterior orbit ESE. Let SK be the mean distance of the bodies P and

S ; and let the accelerative attraction of the body P towards S, at that

mean distance, be expressed by that line SK. Make SL to SK as the

E C

square of SK to the square of SP, and SL will be the accelerative attrac
tion of the body P towards S at any distance SP. Join PT, and draw
LM parallel to it meeting ST in M; and the attraction SL will be resolv
ed (by Cor. 2. of the Laws of Motion) into the attractions SM, LM. And
so the body P will be urged with a threefold accelerative force. One of
these forces tends towards T, and arises from the mutual attraction of the
bodies T and P. By this force alone the body P would describe round the
body T, by the radius PT, areas proportional to the times, and an
ellipsis whose focus is in the centre of the body T ; and this it would do
whether the body T remained unmoved, or whether it were agitated by that
attraction. This appears from Prop. XI, and Cor. 2 and 3 of Theor.
XXI. The other force is that of the attraction LM, which, because it
tends from P to T, will be superadded to and coincide with the former
force ; and cause the areas to be still proportional to the times, by Cor. 3,
Theor. XXI. But because it is not reciprocally proportional to the square
of the distance PT, it will compose, when added to the former, a force
varying from that proportion : which variation will be the greater by how
much the proportion of this force to the former is greater, cceteris paribus.
Therefore, since by Prop. XI, and by Cor. 2, Theor. XXI, the force with
which the ellipsis is described about the focus T ought to be directed to
that focus, and to be reciprocally proportional to the square of the distance
PT, that compounded force varying from that proportion will make the
orbit PAB vary from the figure of an ellipsis that has its focus in the point
'I1 ; and so much the more by how much the variation from that proportion
is greater ; and by consequence by how much the proportion of the second
force LM to the first force is greater, cceteris paribus. But now the third
force SM, attracting the body P in a direction parallel to ST, composes with
the other forces a new force which is no longer directed from P to T : and which
varies so much more from this direction by how much the proportion of this
third force to the other forces is greater, cceteris paribus ; arid therefore causes
the body P to describe, by the radius TP, areas no longer proportional to the
times : and therefore makes the variation from that proportionality so much
greater by how much the proportion of this force to the others is greater.
But this third force will increase the variation of the orbit PAB from th*

THE MATHEMATICAL PRINCIPLES [BOOK 1

elliptical figure before-mentioned upon two accounts ; first because that
force is not directed from P to T ; and, secondly, because it is not recipro
cally proportional to the square of the distance PT. These things being
premised, it is manifest that the areas are then most nearly proportional to
the times, when that third force is the least possible, the rest preserving
their former quantity ; and that the orbit PAB does then approach nearest
to the elliptical figure above-mentioned, when both the second and third,
but especially the third force, is the least possible; the first force remain
ing in its former quantity.

Let the accelerative attraction of the body T towards S be expressed by
the line SN ; then if the accelerative attractions SM and SN were equal,
these, attracting the bodies T and P equally and in parallel directions
would not at all change their situation with respect to each other. The mo
tions of the bodies between themselves would be the same in that case as if
those attractions did not act at all, by Cor. 6, of the Laws of Motion. And,
by a like reasoning, if the attraction SN is less than the attraction SM, it
will take away out of the attraction SM the part SN, so that there will re
main only the part (of the attraction) MN to disturb the proportionality of
the areas and times, and the elliptical figure of the orbit. And in like
manner if the attraction SN be greater than the attraction SM, the pertur
bation of the orbit and proportion will be produced by the difference MN
alone. After this manner the attraction SN reduces always the attraction
SM to the attraction MN, the first and second attractions rema ning per
fectly unchanged ; and therefore the areas and times come then nearest to
proportionality, and the orbit PAB to the above-mentioned elliptical figure,
when the attraction MN is either none, or the least that is possible; that
is, when the accelerative attractions of the bodies P and T approach as near
as possible to equality ; that is, when the attraction SN is neither none at
all, nor less than the least of all the attractions SM, but is, as it were, a
mean between the greatest and least of all those attractions SM, that is,
not much greater nor much less than the attraction SK. Q.E.D.

CASE 2. Let now the lesser bodies P. S, revolve about a greater T in dif
ferent planes ; and the force LM, acting in the direction of the line PT
situate in the plane of the orbit PAB, will have the same effect as before ;
neither will it draw the body P from the plane of its orbit. But the other
force NM acting in the direction of a line parallel to ST (and which, there
fore, when the body S is without the line of the nodes is inclined to the
plane of the orbit PAB), besides the perturbation of the motion just now
spoken of as to longitude, introduces another perturbation also as to latitude,
attracting the body P out of the plane of its orbit. And this perturbation,
in any given situation of the bodies P and T to each other, will be as the
generating force MN ; and therefore becomes least when the force MN is
least, that is (as was just now shewn), where the attraction SN is not nrirb
greater nor much less than the attraction SK. Q.E.D.

SK-C. XL] OF NATURAL PHILOSOPHY. 205

COR. 1. Hence it may be easily collected, that if several less bodies P
8, R, &c.; revolve about a very great body T, the motion of the innermost
revolving body P will be least disturbed by the attractions of the others.
when the great body is as well attracted and agitated by the rest (accord
ing to the ratio of the accelerative forces) as the rest are by each other
mutually.

COR. 2. In a system of three bodies, T, P, S, if the accelerative attrac
tions of any two of them towards a third be to each other reciprocally as the
squares of the distances, the body P, by the radius PT, will describe its area
about the body T swifter near the conjunction A and the opposition B than it
will near the quadratures C arid D. For every force with which the body P
is acted on and the body T is not, and which does not act in the direction of
the line PT, does either accelerate or retard the description of the area,
according as it is directed, whether in consequentia or in cwtecedentia.
Such is the force NM. This force in the passage of the body P frcm C
to A is directed in consequentia to its motion, and therefore accelerates
it; then as far as D in atttecedentia, and retards the motion; then in, con
sequentia as far as B ; and lastly in antecedentia as it moves from B to C.

COR. 3. And from the same reasoning it appears that the body P ccBteris
paribuSj moves more swiftly in the conjunction and opposition than in the
quadratures.

COR. 4. The orbit of the body P, cc&teris paribus, is more curve at the
quadratures than at the conjunction and opposition. For the swifter
bodies move, the less they deflect from a rectilinear path. And besides the
force KL, or NM, at the conjunction and opposition, is contrary to the
force with which the body T attracts the body P, and therefore diminishes
that force ; but the body P will deflect the less from a rectilinear path the
less it is impelled towards the body T.

COR. 5. Hence the body P, cceteris paribus, goes farther from the body
T at the quadratures than at the conjunction and opposition. This is said,
E C_ L

B

however, supposing no regard had to the motion of eccentricity. For if
the orbit of the body P be eccentrical, its eccentricity (as will be shewn
presently by Cor. 9) will be greatest when the apsides are in the syzy-
gies; and thence it may sometimes come to pass that the body P. in its
near approach to the farther apsis, may go farther from the body T at the
syzygies than at the quadratures.

COR. 6. Because the centripetal force of the central body T, by which

206 THE MATHEMATICAL PRINCIPLES [BOOK. 1

the body P is retained in its orbit, is increased at the quadratures by tho
addition caused by the force LM, and diminished at the syzygies by the
subduction caused by the force KL, and, because the force KL is greater
than LM, it is more diminished than increased ; and, moreover, since that
centripetal force (by Cor. 2, Prop. IV) is in a ratio compounded of the sim
ple ratio of the radius TP directly, and the duplicate ratio of the periodi
cal time inversely ; it is plain that this compounded ratio is diminished by
the action of the force KL ; and therefore that the periodical time, supposing
the radius of the orbit PT to remain the same, will be increased, and that
in the subduplicate of that ratio in which the centripetal force is diminish
ed ; and, therefore, supposing this radius increased or diminished, the peri
odical time will be increased more or diminished less than in the sesquipli-
cate ratio of this radius, by Cor. 6, Prop. IV. If that force of the central
body should gradually decay, the body P being less and less attracted would
go farther and farther from the centre T ; and, on the contrary, if it were
increased, it would draw nearer to it. Therefore if the action of the distant
body S, by which that force is diminished, were to increase and decrease
by turns, the radius TP will be also increased and diminshed by turns ;
and the periodical time will be increased and diminished in a ratio com
pounded of the sesquiplicate ratio of the radius, and of the subduplicate oi
that ratio in which the centripetal force of the central body T is dimin
ished or increased, by the increase or decrease of the action of the distant
body S.

COR. 7. It also follows, from what was before laid down, that the axis
of the ellipsis described by the body P, or the line of the apsides, does as
to its angular motion go forwards and backwards by turns, but more for
wards than backwards, and by the excess of its direct motion is in the
whole carried forwards. For the force with which the body P is urged to
the body T at the quadratures, where the force MN vanishes, is compound
ed of the force LM and the centripetal force with which the body T at
tracts the body P. The first force LM, if the distance PT be increased, is
increased in nearly the same proportion with that distance, and the other
force decreases in the duplicate ratio of the distance ; and therefore the
sum of these two forces decreases in a less than the duplicate ratio of the
distance PT ; and therefore, by Cor. 1, Prop. XLV, will make the line of
the apsides, or, which is the same thing, the upper apsis, to go backward.
But at the conjunction and opposition the force with which the body P is
urged towards the body T is the difference of the force KL, and of the
force with which the body T attracts the body P ; and that difference, be
cause the force KL is very nearly increased in the ratio of the distance
PT, decreases in more -than the duplicate ratio of the distance PT ; and
therefore, by' Cor. 1, Prop. XLV, causes the line of the apsides to go for
wards. In the places between the syzygies and the quadratures, the motion

SEC. Xl.J OF NATURAL PHILOSOPHY. 207

of the line of the apsides depends upon both < f these causes conjuncdy, so
that it either goes forwards or backwards in proportion to the excess ol
one of these causes above the other. Therefore since the force KL in the
syzygies is almost twice as great as the force LM in the quadratures, the
excess will be on the side of the force KL, and by consequence the line of
the apsides will be carried forwards. The truth of this arid the foregoing

IE

Corollary will be more easily understood by conceiving the system of the
two bodies T and P to be surrounded on every side by several bodies S,
S, S, dec., disposed about the orbit ESE. For by the actions of these bo
dies the action of the body T will be diminished on every side, and decrease
in more than a duplicate ratio of the distance.

COR. 8. IJut since the progress or regress of the apsides depends upon
the decrease of the centripetal force, that is, upon its being in a greater or
less ratio than the duplicate ratio of the distance TP, in the passage of
the body from the lower apsis to the upper ; and upon a like increase in
its return to the lower apsis again ; and therefore becomes greatest where
the proportion of the force at the upper apsis to the force at the lower ap
sis recedes farthest from the duplicate ratio of the distances inversely ; it
is plain, that, when the apsides are in the syzygies, they will, by reason of
the subducting force KL or NM — LM, go forward more swiftly ; and in
the quadratures by the additional force LM go backward more slowly.
Because the velocity of the progress or slowness of the regress is continued
for a long time ; this inequality becomes exceedingly great.

COR. 9. If a body is obliged, by a force reciprocally proportional to the
square of its distance from any centre, to revolve in an ellipsis round that
centre ; and afterwards in its descent from the upper apsis to the lower
apsis, that force by a perpetual accession of new force is increased in more
than a duplicate ratio of the diminished distance ; it is manifest that the
body, being impelled always towards the centre by the perpetual accession
of this new force, will incline more towards that centre than if it were
urged by that force alone which decreases in a duplicate ratio of the di
minished distance, and therefore will describe an orbit interior to that
elliptical orbit, and at the lower apsis approaching nearer to the centre
than before. Therefore the orbit by the accession of this new force will
become more eccentrical. If now, while the body is returning from the
lower to the upper apsis, it should decrease by the same degrees by which
it increases before the body would return to its first distance; and there-

THE MATHEMATICAL PRINCIPLES [BOOK I.

fore if the force decreases in a yet greater ratio, the body, being now less
attracted than before, will ascend to a still greater distance, and so the ec
centricity of the orbit will be increased still more. Therefore if the ratio
of the increase and decrease of the centripetal force be augmented each
revolution, the eccentricity will be augmented also ; and, on the contrary,
if that ratio decrease, it will be diminished.

Now, therefore, in the system of the bodies T, P, S, when the apsides of
the orbit FAB are in the quadratures, the ratio of that increase and de
crease is least of all, and becomes greatest when the apsides are in the
syzygies. If the apsides are placed in the quadratures, the ratio near the
apsides is less, and near the syzygies greater, than the duplicate ratio of the
distances : and from that Greater ratio arises a direct motion of the line of

7 o

the apsides, as was just now said. But if we consider the ratio of the
whole increase or decrease in the progress between the apsides, this is less
than the duplicate ratio of the distances. The force in the lower is to the
force in the upper apsis in less than a duplicate ratio of the distance of the
upper apsis from the focus of the ellipsis to the distance of the lower apsis
from the same focus ; and, contrariwise, when the apsides are placed in the
syzygies, the force in the lower apsis is to the force in the upper apsis in a
greater than a duplicate ratio of the distances. For the forces LM in the
quadratures added to the forces of the body T compose forces in a less ra
tio ; and the forces KL in the syzygies subducted from the forces of the
body T, leave the forces in a greater ratio. Therefore the ratio of the
whole increase and decrease in the passage between the apsides is least at
the quadratures and greatest at the syzygies ; and therefore in the passage
of the apsides from the quadratures to the syzygies it is continually aug
mented, and increases the eccentricity of the ellipsis ; and in the passage
from the syzygies to the quadratures it is perpetually decreasing, and di
minishes the eccentricity.

COR. 10. That we may give an account of the errors as to latitude, let
us suppose the plane of the orbit EST to remain immovable; and from
the cause of the errors above explained, it is manifest, that, of the two
forces NM, ML, which are the only and entire cause of them, the force
ML acting always in the plane of the orbit PAB never disturbs the mo
tions as to latitude ; and that the force NM, when the nodes are in the
gyzygies, acting also in the same plane of the orbit, does not at that time
affect those motions. But when the nodes are in the quadratures, it dis
turbs tliem very much, and, attracting the body P perpetually out of the
plane of its orbit, it diminishes the inclination of the plane in the passage
of the body from the quadratures to the syzygies, and again increases the
same in the passage from the syzygies to the quadratures. Hence it
comes to pass that when the body is in the syzygies, the inclination is
then least of all, and returns to the first magnitude nearly, when the body

SEC. XL] OF NATURAL PHILOSOPHY. 209

arrives at the next node. But if the nodes are situate at the octants after
the quadratures, that is, between C and A, D and B, it will appear, from

ii C L

E

wnat was just now shewn, that in the passage of the body P from either
node to the ninetieth degree from thence, the inclination of the plane is
perpetually diminished ; then, in the passage through the next 45 degrees
to the next quadrature, the inclination is increased ; and afterwards, again,
in its passage through another 45 degrees to the next node, it is dimin
ished. Therefore the inclination is more diminished than increased, and
is therefore always less in the subsequent node than in the preceding one.
And, by a like reasoning, the inclination is more increased than diminish
ed when the nodes are in the other octants between A and D, B and C.
The inclination, therefore, is the greatest of all when the nodes are in the
syzygies In their passage from the syzygies to the quadratures the incli
nation is diminished at each appulse of the body to the nodes : and be
comes least of all when the nodes are in the quadratures, and the body in
the syzygies ; then it increases by the same degrees by which it decreased
before ; and, when the nodes come to the next syzygies, returns to its
former magnitude.

COR. 11. Because when the nodes are in the quadratures the body P is
perpetually attracted from the plane of its orbit ; and because this attrac
tion is made towards S in its passage from the node C through the con
junction A to the node D ; and to the contrary part in its passage from the
node D through the opposition B to the node C; it is manifest that, in its
motion from the node C, the body recedes continually from the former
plane CD of its orbit till it comes to the next node; and therefore at that
node, being now at its greatest distance from the first plane CD, it will
pass through the plane of the orbit EST not in D, the other node of that
plane, but in a point that lies nearer to the body S, which therefore be
comes a new place of the node in, antecedent ia to its former place. And,
by a like reasoning, the nodes will continue to recede in their passage
from this node to the next. The nodes, therefore, when situate in the
quadratures, recede perpetually ; and at the syzygies, where no perturba
tion can be produced in the motion as to latitude, are quiescent : in the in
termediate places they partake of both conditions, and recede more slowly ;
and, therefore, being always either retrograde or stationary, they will be
carried backwards, or in atitecedentia, each revolution.

COR. 12. All the errors described in these corrollaries arc a little greater

14

210 THE MATHEMATICAL PRINCIPLES BOOK L

at the conjunction of the bodies P, S, than at their opposition ; because
the generating forces NM and ML are greater.

COR. 13. And since the causes and proportions of the errors and varia
tions mentioned in these Corollaries do not depend upon the magnitude of
the body S, it follows that all things before demonstrated will happen, if
the magnitude of the body S be imagined so great as that the system of the
two bodies P and T may revolve about it. And from this increase of the
body S, and the consequent increase of its centripetal force, from which the
errors of the body P arise, it will follow that all these errors, at equal dis
tances, will be greater in this case, than in the other where the body S re
volves about the system of the bodies P and T.

COR. 14. But since the forces NM, ML, when the body S is exceedingly
distant, are very nearly as the force SK and the ratio PT to ST con-
junctly ; that is, if both the distance PT, and the absolute force of the body
8 be given, as ST3 reciprocally : and since those forces NM, ML are the
causes of all the errors and effects treated of in the foregoing Corollaries;
it is manifest that all those effects, if the system of bodies T and P con
tinue as before, and only the distance ST and the absolute force of the body
S be changed, will be very nearly in a ratio compounded of the direct ratio
of the absolute force of the body S, and the triplicate inverse ratio of the
distance ST. Hence if the system of bodies T and P revolve about a dis
tant body S, those forces NM, ML, and their eifl ts, will be (by Cor. 2 and
6, Prop IV) reciprocally in a duplicate ratio c/f the periodical time. And
thence, also, if the magnitude of the bodv S be proportional to its absolute
force, those forces NM, ML, and their effects, will be directly as the cube
of the apparent diameter of the distant body S viewed from T, and so vice
versa. For these ratios are the same as the compounded ratio above men
tioned.

COR. 15. And because if the orbits ESE and PAB, retaining their fig
ure, proportions, and inclination to each other, should alter their magni
tude ; arid the forces of the bodies S and T should either remain, or be
changed in any given ratio ; these forces (that is, the force of the body T,
which obliges the body P to deflect from a rectilinear course into the orbit
PAB, and the force of the body S, which causes the body P to deviate from
that orbit) would act always in the same manner, and in the same propor
tion : it follows, that all the effects will be similar and proportional, arid
the times of those effects proportional also ; that is, that all the linear er
rors will be as tne diameters of the orbits, the angular errors the same as
before ; and the times of similar linear errors, or equal angular errors? as
the periodical times of the orbits.

COR. 16. Therefore if the figures of the orbits and their inclination to
each other be given, and the magnitudes, forces, arid distances of the bodies
he any how changed, we may. from the errors and times of those errors in

SEC. XI.] OF NATURAL PHILOSOPHY. 2 \\

one case, collect very nearly the errors and times of the errors in any other
case. But this may be done more expeditiously by the following method.
The forces NM; ML, other things remaining unaltered, are as the radius
TP ; and their periodical effects (by Cor. 2, Lein. X) are as the forces and
the square of the periodical time of the body P conjunctly. These are the
linear errors of the body P ; and hence the angular errors as they appear
from the centre T (that is, the motion of the apsides and of the nodes, and all
the apparent errors as to longitude and latitude) are in each revolution of
the body P as the square of the time of the revolution, very nearly. Let
these ratios be compounded with the ratios in Cor. 14, and in any system
of bodies T, P, S, where P revolves about T very near to it, and T re
volves about S at a great distance, the angular errors of the body P, ob
served from the centre T, will be in each revolution of the body P as the
square of the periodical time of the body P directly, and the square of the
periodical time of the body T inversely. And therefore the mean motion
of the line of the apsides will be in a given ratio to the mean motion of
the nodes ; and both those motions will be as the periodical time of the
body P directly, and the square of the periodical time of the body T in
versely. The increase or diminution of the eccentricity and inclination of
the orbit PAB makes no sensible variation in the motions of the apsides*
and nodes, unless that inc/case or diminution be very great indeed.

COR. 17. Sines the line LM becomes sometimes greater and sometimes
less than the radius PT, let the mean quantity of the force LM be expressed
E C

sa— - -•::-.•.•::::::;

by that radius PT ; and then that mean force will be to the mean force
SK or SN (which may be also expressed by ST) as the length PT to the
length ST. But the mean force SN or ST, by which the body T is re
tained in the orbit it describes about S, is to the force with which the body P
is retained in its orbit about T in a ratio compounded of the ratio of the
radius ST to the radius PT, and the duplicate ratio of the periodical time
of the body P about T to the periodical time of the body T about S. And,
ex cequo, the mean force LM is to the force by which the body P is retain
ed in its orbit about T (or by which the same body P might revolve at the
distance PT in the same periodical time about any immovable point T) in
the same duplicate ratio of the periodical times. The periodical times
therefore being given, together with the distance PT, the mean force LM
is also given ; and that force being given, there is given also the force MN,
very nearly, by the analogy of the lines PT and MN.

212 THE MATHEMATICAL PRINCIPLES [BoOK I

Con. IS. By tlie same laws by which the body P revolves about the
body T, let us suppose many fluid bodies to move round T at equal dis
tances from it ; and to be so numerous, that they may all become contiguous
to each other, so as to form a fluid annul us, or ring, of a round figure, and
concentrical to the body T; and the several parts of this annulus, perform
ing their motions by the same law as the body P, will draw nearer to the
body T, and move swifter in the conjunction and opposition of themselves
and the body S, than in the quadratures. And the nodes of this annulus,
or its intersections with the plane of the orbit of the body S or T, will rest
at the syzygies ; but out of the syzygies they will be carried backward, or
in. antecedentia ; with the greatest swiftness in the quadratures, and more
slowly in other places. The inclination of this annulus also will vary, and
its axis will oscillate each revolution, and when the revolution is completed
will return to its former situation, except only that it will be carried round
a little by the precession of the nodes.

COR. 19. Suppose now the spherical body T, consisting of some matter
not fluid, to be enlarged, and to extend its 'If on every side as far as that
annulus, and that a channel were cut all round its circumference contain
ing water j and that this sphere revolves uniformly about its own axis in
the same periodical time. This water being accelerated and retarded by
turns (as in the last Corollary), will be swifter at the syzygies, and slower
at the quadratures, than the surface of the globe, and so will ebb and flow in
its channel after the manner of the sea. If the attraction of the body S were
taken away, the water would acquire no motion of flux and reflux by revolv-
.ng round the quiescent centre of the globe. The case is the same of a globe
moving uniformly forwards in a right line, and in the mean time revolving
about its centre (by Cor. 5 of the Laws of Motion), and of a globe uni
formly attracted from its rectilinear course (by Cor. 6, of the same Laws).
But let the body S come to act upon it, and by its unequable attraction the
A\ater will receive this new motion ; for there will be a stronger attraction
upon that part of the water that is nearest to the body, and a weaker upon
that part which is more remote. And the force LM will attract the water
downwards at the quadratures, and depress it as far as the syzygies ; and the
force KL will attract it upwards in the syzygies, and withhold its descent,
and make it rise as far as the quadratures ; except only in so far as the
motion of flux and reflux may be directed by the channel of the water, and
be a little retarded by friction.

COR. 20. If, now, the annulus becomes hard, and the globe is diminished,
the motion of flux and reflux will cease ; but the oscillating motion of the
inclination and the praecession of the nodes will remain. Let the globe
have the same axis with the annulus, and perform its revolutions in the
same times, and at its surface touch the annulus within, and adhere to it;
then the globe partaking of the motion of the annulus, this whole compares

SEC. XI. OF NATURAL PHILOSOPHY. 213

will oscillate, and the nodes will go backward, for the globe, as \ve shall
shew presently, is perfectly indifferent to the receiving of all impressions.
The greatest angle of the inclination of the annulus single is when the
nodes are in the syzygies. Thence in the progress of the nodes to the
quadratures, it endeavours to diminish its inclination, and by that endea
vour impresses a motion upon the whole globe. The globe retains this
motion impressed, till the annulus by a contrary endeavour destroys that
motion, and impresses a new motion in a contrary direction. And by this
means the greatest motion of the decreasing inclination happens when the
nodes are in the quadratures; and the least angle of inclination in the octants

B

after the quadratures ; and, again, the greatest motion of roclination happens
when the nodes are in the syzygies ; and the greatest angle of reclination in
the octants following. And the case is the same of a globe without this an
nulus, if it be a little higher or a little denser in the equatorial than in the
polar regions : for the excess of that matter in the regions near the equator
supplies the place of the annulus. And though we should suppose the cen
tripetal force of this globe to be any how increased, so that all its parts
were to tend downwards, as the parts of our earth gravitate to the centre,
yet the phenomena of this and the preceding Corollary would scarce be al
tered ; except that the places of the greatest and least height of the water
will be different : for the water is now no longer sustained and kept in its
orbit by its centrifugal force, but by the channel in which it flows. And,
besides, the force LM attracts the water downwards most in the quadra
tures, and the force KL or NM — LM attracts it upwards most in the
syzygies. And these forces conjoined cease to attract the water downwards,
and begin to attract it upwards in the octants before the syzygies ; and
cease to attract the water upwards, and begin to attract the water down
wards in the octants after the syzygies. And thence the greatest height of
the water may happen about the octants after the syzygies ; and the least
height about the octants after the quadratures ; excepting only so far as the
motion of ascent or descent impressed by these forces may by the vis insita
of the water continue a little longer, or be stopped a little sooner by impe
diments in its channel.

COR. 21. For the same reason that redundant matter in the equatorial
regions of a globe causes the nodes to go backwards, and therefore by the
increase of that matter that retrogradation is increased, by the diminution
is diminished, and by the removal quite ceases : it follows, that, if more than

214 THE MATHEMATICAL PRINCIPLES [BOOK I

that redundant matter be taken away, that is, if the globe be either more
depressed, or of a more rare consistence near the equator than near the
poles, there will arise a motion of the nodes in consequentia.

COR. 22. And thence from the motion of the nodes is known the consti
tution of the globe. That is, if the globe retains unalterably the same poles,
and the motion (of the nodes) be in. antecedetitia, there is a redundance oi
the matter near the equator; but if in conseqnentia, a deficiency. Sup
pose a uniform and exactly spherical globe to be first at rest in a free space :
then by some impulse made obliquely upon its superficies to be driven from
its place, and to receive a motion partly circular and partly right forward.
Because this globe is perfectly indifferent to all the axes that pass through
its centre, nor has a greater propensity to one axis or to one situation oi
the axis than to any other, it is manifest that by its own force it will never
change its axis, or the inclination of it. Let now this globe be impelled
obliquely by a new impulse in the same part of its superficies as before .
and since the effect of an impulse is not at all changed by its coming sooner
or later, it is manifest that these two impulses, successively impressed, will
produce the same motion as if they were impressed at the same time : that
is, the same motion as if the globe had been impelled by a simple force
compounded of them both (by Cor. 2, of the Laws), that is, a simple motion
about an axis of a given inclination. And the case is the same if the sec
ond impulse were made upon any other place of the equator of the first
motion ; and also if the first impulse were made upon any place in the
equator of the motion which would be generated by the second impulse
alone; and therefore, also, when both impulses are made in any places
whatsoever ; for these impulses will generate the same circular motion as
if they were impressed together, and at once, in the place of the intersec
tions of the equators of those motions, which would be generated by each
of them separately. Therefore, a homogeneous and perfect globe will not
retain several distinct motions, but will unite all those that are impressed
on it, and reduce them into one; revolving, as far as in it lies, always with
a simple and uniform motion about one single given axis, with an inclina
tion perpetually invariable. And the inclination of the axis, or the velocity
of the rotation, will not be changed by centripetal force. For if the globe
be supposed to be divided into two hemispheres, by any plane whatsoever
passing through its own centre, and the centre to which the force is direct
ed, that force will always urge each hemisphere equally ; and therefore will
not incline the globe any way as to its motion round its own axis. But
let there be added any where between the pole and the equator a heap oi
new matter like a mountain, and this, by its perpetual endeavour to recede
from the centre of its motion, will disturb the motion of the globe, and
cause its poles to wander about its superficies, describing circles about
themselves and their opposite points. Neither can this enormous evagatior

XL] OF NATURAL PHILOSOPHY. 2 In

of the poles be corrected, unless by placing that mountain ei '. er in one ol
the poles; in which case, by Cor. 21, the nodes of the equator will go for
wards ; or in the equatorial regions, in which case, by Cor. 20, the nodes
will go backwards: or, lastly, by adding on the other side of the axis anew
quantity of matter, by which the mountain may be balanced in its motion;
and then the nodes will either go forwards or backwards, as the mountain
and this newly added matter happen to be nearer to the pole or to the
equator.

PROPOSITION LXV1I. THEOREM XXVII.

The same laics of attraction being' supposed, I say, that the exterior body
S does, by radii dra.cn to the point O, the common centre of gravity
of the interior bodies P and T, describe round that centre areas more
proportional to the times, and an orbit more approaching to the form
of an ellipsis having its focus in that cen > •.-. than, it can describe
round the innermost and greatest body T by ra Hi drawn to that
body.
For the attractions of the body S towards T and

P compose its absolute attraction, which is more

directed towards O, the common centre of gravity S(i

of the bodies T and P, than it is to the «. reatest

body T ; and which is more in a reciprocal propor
tion to the square of the distance SO, than it is to the square of the distance

ST : as will easily appear by a little consideration.

PROPOSITION LXVIII. THEOREM XXVIII.

The same laws of attraction supposed, I say, that the exterior body S
will, by radii drawn to O, the common centre of gravity of the interior
bodies P and T, describe round that centre areas more propor
tional to the times, and an orbit more approaching to the form of an
ellipsis having its focus in that centre, if the innermost and greatest
body be agitated by these attractions as well as the rest, than it would
do if that body were either at rest as not attracted, or were much tnore
or much less attracted, or much more or much less agitated.
This may be demonstrated after the same manner as Prop. LXVI, but
by a more prolix reasoning, which I therefore pass over. It will be suf
ficient to consider it after this manner. From the demonstration of the
last Proposition it is plain, that the centre, towards which the body S is
urged by the two forces conjunctly, is very near to the common centre of
gravity of those two other bodies. If this centre were to coincide with that
common centre, and moreover the common centre of gravity of all the three
bodies were at rest, the body S on one side, and the common centre of
gravity of the other two bodies on the other side, would describe true ellip*

216 THE MATHEMATICAL PRINCIPLES - [BOOK 1

ses about that quiescent common centre. This appears from Cor. 2, Pro])
LVIII, compared with what was demonstrated in Prop. LX1V, and LXY
Now this accurate elliptical motion will be disturbed a little by the dis
tance of the centre of the two bodies from the centre towards which tht
third body S is attracted. Let there be added, moreover, a motion to the
Bommon centre of the three, and the perturbation will be increased yet
more. Therefore the perturbation is least when the
common centre of the three bodies is at rest; that
I is, when the innermost and greatest body T is at
tracted according to the same law as the rest are ;
and is always greatest when the common centre of
the three, by the diminution of the motion of the body T, begins to be
moved, and is more and more agitated.

COR. And hence if more lesser bodies revolve about the great one, it
may easily be inferred that the orbits described will approach nearer to
ellipses ; and the descriptions of areas will be more nearly equable, if all
the bodies mutually attract and agitate each other with accelerative forces
that are as their absolute forces directly, and the squares of the distances
inversely : and if the focus of each orbit be placed in the common centre
of gravity of all the interior bodies (that is. if the focus of the first and in
nermost orbit be placed in the centre of gravity of the greatest and inner
most body : the focus of the second orbit in the common centre of gravity
of the two innermost bodies; the focus of the third orbit in the common
centre of gravity of the three innermost ; and so on), than if the innermost
body were at rest, and was made the common focus of all the orbits.

PROPOSITION LXIX. THEOREM XXIX.

fn a system of several bodies A, B, C, D, $*c., if any one of those bodies,
as A, attract all the rest, B, C, D, $*c.,with accelerative forces that are
reciprocally as the squares of the distances from the attracting body ;
and another body, as B, attracts also the rest. A, C, D, $-c., with forces
that are reciprocally as the squares of the distances from the attract
ing body ; the absolute forces of the attracting bodies A and B will
be to each other as those very bodies A and B to which those forces
belong.

For the accelerative attractions of all the bodies B, C, D, towards A,
are by the supposition equal to each other at equal distances ; and in like
manner the accelerative attractions of all the bodies towards B are also
equal to each other at equal distances. But the absolute attractive force
of the body A is to the absolute attractive force of the body B as the ac-
eelerative attraction of all the bodies towards A to the accelerative attrac
tion of all the bodies towards B at equal distances ; and so is also the ac
celerative attraction of the body B to*vards A to the accelerative attraction

SEC. XI] OF NATURAL PHILOSOPHY. 21 T

of the body A towards B. But the accelerative attraction of the body B
towards A is to the accelerative attraction of the body A towards B as the
mass of the body A to the mass of the body B ; because the motive forces
which (by the 2d, 7th, and 8th Definition) are as the accelerative forces
and the bodies attracted conjunctly are here equal to one another by the
third Law. Therefore the absolute attractive force of the body A is to the
absolute attractive force of the body B aa the mass of the body A to the
mass of the body B. Q.E.D.

COR. 1. Therefore if each of the bodies of the system A, B, C, D, &c.
does singly attract all the rest with accelerative forces that are reciprocally
as the squares of the distances from the attracting body, the absolute forces
of all those bodies will be to each other as the bodies themselves.

COR. 2. By a like reasoning, if each of the bodies of the system A, B,
C, D, &c., do singly attract all the rest with accelerative forces, which are
either reciprocally or directly in the ratio of any power whatever of the
distances from the attracting body : or which are defined by the distances
from each of the attracting bodies according to any common law : it is plain
that the absolute forces of those bodies are as the bodies themselves.

COR. 3. In a system of bodies whose forces decrease in the duplicate ra
tio of the distances, if the lesser revolve about one very great one in ellip
ses, having their common focus in the centre of that great body, and of a
figure exceedingly accurate ; and moreover by radii drawn to that great
ody describe areas proportional to the times exactly • the absolute forces
)i those bodies to each other will be either accurately or very nearly in the
ratio of the bodies. And s > on the contrary. This appears from Cor. of
Prop. XLVII1, compared with the first Corollary of this Prop.

SCHOLIUM.

These Propositions naturally lead us to the analogy there is between
centripetal forces, and the central bodies to which those forces used to be
directed ; for it is reasonable to suppose that forces which are directed to
bodies should depend upon the nature and quantity of those bodies, as we
see they do in magnetical experiments. And when such cases occur, we
are to compute the attractions of the bodies by assigning to each of their
particles its proper force, and then collecting the sum of them all. I here
ue*e the word attraction in general for any endeavour, of what kind soever,
made by bodies to approach to each other; whether that endeavour arise
from the action of the bodies themselves, as tending mutually to or agita
ting each other by spirits emitted; or whether it arises from the action
of the aether or of the air, or of any medium whatsoever* whether corporeal
or incorporeal, any how impelling bodies placed therein towards each other.
In the same general sense I use the word impulse, not defining in this trea
tise the species or physical qualities of forces, but investigating the quantities

THE MATHEMATICAL PRINCIPLES [BOOK ).

and mathematical proportions of them ; as I observed before ir (lie Defi
nitions. In mathematics we are to investigate the quantities of forces
with their proportions consequent upon any conditions supposed ; then,
when we enter upon physics, we compare those proportions with the phe
nomena of Nature, that we may know what conditions of those forces an
swer to the several kinds of attractive bodies. And this preparation being
made, we argue more safely concerning the physical species, causes, and
proportions of the forces. Let us see, then, with what forces spherical
bodies consisting of particles endued with attractive powers in the manner
above spoken of must act mutually upon one another : and what kind of
motions will follow from thence.

SECTION XII.

Of the attractive forces of sphcerical bodies.

PROPOSITION LXX. THEOREM XXX.

If to every point of a spherical surface there tend equal centripetal forces
decreasing in, the duplicate ratio of the distances from those points ;
I say, that a corpuscle placed within that superficies will not be attract
ed by those forces any way.

Let HIKL, be that sphaerical superficies, and P a
corpuscle placed within. Through P let there be
drawn to this superficies to two lines HK, IL, inter-
cepting very small arcs HI, KL ; and because (by
Cor. 3, Lem. VII) the triangles HPI,LPK are alike,
those arcs will be proportional to the distances HP
LP ; and any particles at HI and KL of the spheri
cal superficies, terminated by right lines passing through P, will be in the
duplicate ratio of those distances. Therefore the forces of these particles
exerted upon the body P are equal between themselves. For the forces are
as the particles directly, and the squares of the distances inversely. And
these two ratios compose the ratio of equality. The attractions therefore,
being made equally towards contrary parts, destroy each other. And by a
like reasoning all the attractions through the whole spherical superficies
are destroyed by contrary attractions. Therefore the body P will not be
any way impelled by those attractions. Q.E.D.

PROPOSITION LXXI. THEOREM XXXI.

The same things supposed as above, I say, that a cor pu vie placed with
out the sph(eric»l superficies is attracted towards the centre of tht
sphere wiih a force reciprocally proportional to the square of its dis
tance from that centre.
Let AHKB, ahkb, be two equal sphaerical superficies described about

SEC. XII.J OF NATURAL PHILOSOPHY.

the centre S, s ; their diameters AB, ab ; and let P and p be two corpus
cles situate without the gpheres in those diameters produced. Let there

be drawn from the corpuscles the lines PHK, PIL, phk, pil, cutting off
from the great circles AHB, ahb, the equal arcs HK, hk, IL; il ; and to
those lines let fall the perpendiculars SD, sd, SE, SP, 1R, ir ; of which let
SD, sd, cut PL, pi, in F and f. Let fall also to the diameters the perpen
diculars IQ, iq. Let now the angles DPE, dpe, vanish; and because DS
and ds, ES and es are equal, the lines PE, PP, and pe, pf, and the lineolso
I )F, df may be taken for equal ; because their last ratio, when the angles
DPE, dpe vanish together, is the ratio of equality. These things then
supposed, it will be, as PI to PF so is RI to DF, and as pf to pi so is df or
DF to ri ; and, ex cequo, as PI X pf to PF X pi so is R I to ri, that is
(by Cor. 3, Lem VII), so is the arc IH to the arc ih. Again, PI is to PS
as IQ. to SE, and ps to pi as se or SE to iq ; and, ex ceqno, PI X ps to
PS X pi as IQ. to iq. And compounding the ratios PI2 X pf X ps is to
pi2 X PF X PS, as IH X IQ to ih X iq ; that is, as the circular super
ficies which is described by the arc IH, as the semi-circle AKB revolves
about the diameter AB, is to the circular superficies described by the arc ih
as the semi-circle akb revolves about the diameter ab. And the forces
with which these superficies attract the corpuscles P and p in the direction
of lines tending to those superficies are by the hypothesis as the superficies
themselves directly, and the squares of the distances of the superficies from
those corpuscles inversely; that is, as pf X ps to PF XPS. And these
forces again are to the oblique parts of them which (by the resolution of
forces as in Cor. 2, of the Laws) tend to the centres in the directions of the
lines PS, JDS-, as PI to PQ, and pi to pq ; that is (because of the like trian
gles PIQ and PSF, piq and psf\ as PS to PF and ps to pf. Thence ex
cequO) the attraction of the corpuscle P towards S is to the attraction of

PF XpfXps. pf X PF X PS .
the corpusclejo towards 5 as ~ =— is to — , that is,

as ps2 to PS2. And, by a like reasoning, the forces with which the su
perficies described by the revolution of the arcs KL, kl attract those cor
puscles, will be as jDS2 to PS2. And in the same ratio will be the foroes
of all the circular superficies into which each of the sphaerical superficies
may be divided by taking sd always equal to SD, and se equal to SE. And
therefore, by composition, the forces of the entire spherical superficies ex
erted upon those corpuscles will be in the same ratio. Q.E.D

220 THE MATHEMATICAL PRINCIPLES [BOOK 1

PROPOSITION LXXIL THEOREM XXXII.

If to the several points of a sphere there tend equal centripetal forces de
creasing in a duplicate ratio of the distances from those points ; and
there be given both the density of the sphere and the ratio of the di
ameter of the sphere to the distance of the corpuscle from its centre ;
I say, that the force with which the corpuscle is attracted is propor
tional to the semi-diameter of the sphere.

For conceive two corpuscles to be severally attracted by two spheres, one
by one, the other by the other, and their distances from the centres of the
spheres to be proportional to the diameters of the spheres respectively , and
the spheres to be resolved into like particles, disposed in a like situation
to the corpuscles. Then the attractions of one corpuscle towards the sev
eral particles of one sphere will be to the attractions of the other towards
as many analogous particles of the other sphere in a ratio compounded of
the ratio of the particles directly, and the duplicate ratio of the distances
inversely. But the particles are as the spheres, that is, in a triplicate ra
tio of the diameters, and the distances are as the diameters ; and the first
ratio directly with the last ratio taken twice inversely, becomes the ratio
of diameter to diameter. Q.E.D.

COR. 1. Hence if corpuscles revolve in circles about spheres composed
of matter equally attracting, and the distances from the centres of the
spheres be proportional to their diameters, the periodic times will be equal.
COR. 2. And, vice versa, if the periodic times are equal, the distances
will be proportional to the diameters. These two Corollaries appear from
Cor. 3, Prop. IV.

COR. 3. If to the several points of an^ two solids whatever, of like fig-
are and equal density, there tend equal centripetal forces decreasing in a
duplicate ratio of the distances from those points, the forces, with which
corpuscles placed in a like situation to those two solids will be attracted
by them, will be to each other as the diameters of the solids.

PROPOSITION LXXIII. THEOREM XXXIII.

If to the several points of a given sphere there tend equal centripetal forces
decreasing in a duplicate ratio of the distances from the points ; 1
say, that a corpuscle placed within the sphere is attracted by a force
proportional to its distance from the centre.

In the sphere ABCD, described about the centre S,
let there be placed the corpuscle P ; and about the
same centre S, with the interval SP? conceive de-
|B scribed an interior sphere PEQP. It is plain (by
Prop. LXX) that the concentric sphaerical superficies,
of which the difference AEBF of the spheres is com
posed, have no effect at all upon the body P, their at-

SEC. XIL] OF NATURAL PHILOSOPHY. 22\

tractions being destroyed by contrary attractions. There remains, there
fore; only the attraction of the interior sphere PEQ,F. And (by Prop.
LXXII) this is as the distance PS. Q.E.D.

SCHOLIUM.

By the superficies of which I here imagine the solids composed, I do not
mean superficies purely mathematical, but orbs so extremely thin, that
their thickness is as nothing; that is, the evanescent orbs of which the sphere
will at last consist when the number of the orbs is increased, and their
thickness diminished without end. In like manner, by the points of which
lines, surfaces, and solids are said to be composed, are to be understood
equal particles, whose magnitude is perfectly inconsiderable.

PROPOSITION LXXIV. THEOREM XXXIV.

The same things supposed, I say, that a corpuscle situate without the

sphere is attracted with a force reciprocally proportional to the square

of its distance from the centre.

For suppose the sphere to be divided into innumerable concentric sphe
rical superficies, and the attractions of the corpuscle arising from the sev
eral superficies will be reciprocally proportional to the square of the dis
tance of the corpuscle from the centre of the sphere (by Prop. LXXI).
And, by composition, the sum of those attractions, that is, the attraction
of the corpuscle towards the entire sphere, will be in the same ratio. Q.E.D.

COR. 1. Hence the attractions of homogeneous spheres at equal distances
from the centres will be as the spheres themselves. For (by Prop. LXXII)
if the distances be proportional to the diameters of the spheres, the forces
will be as the diameters. Let the greater distance be diminished in that
ratio ; and the distances now being equal, the attraction will be increased
in the duplicate of that ratio ; and therefore will be to the other attraction
in the triplicate of that ratio ; that is, in the ratio of the spheres.

COR. 2. At any distances whatever the attractions are as the spheres
applied to the squares of the distances.

COR. 3. If a corpuscle placed without an homogeneous sphere is attract
ed by a force reciprocally proportional to the square of its distance from
the centre, and the sphere consists of attractive particles, the force of ever y
particle will decrease in a duplicate ratio of the distance from each particle.

PROPOSITION LXXV. THEOREM XXXV.

If to the several points of a given sphere there tend equal centripetal forces
decreasing in a duplicate ratio of the distances from the points ; I say,
that another similar sphere will be attracted by it with a force recip
rocally proportional to the square of the distance of the centres.
For the attraction of every particle is reciprocally as the square of its

222 THE MATHEMATICAL PRINCIPLES | BOOK L

distance from the centre of the attracting sphere (by Prop. LXXIV). and
is therefore the same as if that whole attracting force issued from one sin
gle corpuscle placed in the centre of this sphere. But this attraction is as
great as on the other hand the attraction of the same corpuscle would be,
if that were itself attracted by the several particles of the attracted sphere
with the same force with which they are attracted by it. But that attrac
tion of the corpuscle would be (by Prop. LXXIV) reciprocally propor
tional to the square of its distance from the centre of the sphere : therefore
the attraction of the sphere, equal thereto, is also in the same ratio. Q,.E.D.

COR. 1. The attractions of spheres towards other homogeneous spheres
are as the attracting spheres applied to the squares of the distances of their
centres from the centres of those which they attract.

COR. 2. The case is the same when the attracted sphere does also at
tract. For the several points of the one attract the several points of the
other with the same force with which they themselves are attracted by the
others again; and therefore since in all attractions (by Law III) the at
tracted and attracting point are both equally acted on, the force will be
doubled by their mutual attractions, the proportions remaining.

COR. 3. Those several truths demonstrated above concerning the motion
of bodies about the focus of the conic sections will take place when an
attracting sphere is placed in the focus, and the bodies move without the
sphere.

COR. 4. Those things which were demonstrated before of the motion of
bodies about the centre of the conic sections take place when the motions
are performed within the sphere.

PROPOSITION LXXVI. THEOREM XXXVI.

ff spheres be however dissimilar (as to density of matter and attractive,
force] in the same ratio onward from the centre to the circumference ;
but every where similar, at every given distance from the centre, on all
sides round about ; and the attractive force of every point decreases
in the duplicate ratio of the distance of the body attracted ; I say,
that the whole force with which one of these spheres attracts the oilier
will be reciprocally proportional to the square of the distance of the
centres.

Imagine several concentric similar
spheres, AB, CD, EF, &c.. the inner
most of which added to the outermost
may compose a matter more dense to
wards the centre, or subducted from
them may leave the same more lax and
rare. Then, by Prop. LXXV, these
sphere? will attract other similar con-

SEC. XII.] OF NATURAL PHILOSOPHY. 223

eentric spheres GH; IK, LM, &c., each the other, with forces reciprocally
proportional to the square of the distance SP. And, by composition or
division, the sum of all those forces, or the excess of any of them above
the others; that is, the entire force with which the whole sphere AB (com
posed of any concentric spheres or of their differences) will attract the
whole sphere GH (composed of any concentric spheres or their differences)
in the same ratio. Let the number of the concentric spheres be increased
in infinitum, so that the density of the matter together with the attractive
force may, in the progress from the circumference to the centre, increase or
decrease according to any given law ; and by the addition of matter not at
tractive, let the deficient density be supplied, that so the spheres may acquire
any form desired ; and the force with which one of these attracts the other
will be still, by the former reasoning, in the same ratio of the square of the
distance inversely. Q.E.I).

COR. I. Hence if many spheres of this kind, similar in all respects, at
tract each other mutually, the accelerative attractions of each to each, at
any equal distances of the centre's, will be as the attracting spheres.

COR. 2. And at any unequal distances, as the attracting spheres applied
to the squares of the distances between the centres.

/' COR. 3. The motive attractions, or the weights of the spheres towards
one another, will be at equal distances of the centres as the attracting and
attracted spheres conjunctly ; that is, as the products arising from multi
plying the spheres into each other.

COR. 4. And at unequal distances, as those products directly, and the
squares of the distances between the centres inversely.

COR. 5. These proportions take place also when the attraction arises
from the attractive virtue of both spheres mutually exerted upon each
other. For the attraction is only doubled by the conjunction of the forces,
the proportions remaining as before.

COR. 6. If spheres of this kind revolve about others at rest, each about
each ; and the distances between the centres of the quiescent and revolving
bodies are proportional to the diameters of the quiescent bodies ; the peri
odic times will be equal.

COR. 7. And, again, if the periodic times are equal, the distances will
be proportional to the diameters.

COR. 8. All those truths above demonstrated, relating to the motions
'jf bodies about the foci of conic sections, will take place when an attract
ing sphere, of any form and condition like that above described, is placed
in the focus.

COR. 9. And also when the revolving bodies are also attracting spheres
Df any condition like that above described.

224 THE MATHEMATICAL PRINCIPLES [BOOK I.

PROPOSITION LXXVI1. THEOREM XXXVII.

Tf to 1 he several points of spheres there tend centripetal forces propor
tional to the distances of the points from the attracted bodies ; I say,
that the compounded force with which two spheres attract each other
mutually is as the distance between the centres of the spheres.

CASE 1. Let AEBF be a sphere ; S its
centre . P a corpuscle attracted : PA SB
the axis of the sphere passing through the
centre of the corpuscle ; EF, ef two planes
cutting the sphere, and perpendicular to
the axis, and equi-distant, one on one side,
the other on the other, from the centre of
the sphere ; G and g- the intersections of
the planes and the axis ; and H any point in the plane EF. The centri
petal force of the point H upon the corpuscle P, exerted in the direction of
the line PH, is as the distance PH ; and (by Cor. 2, of the Laws) the same
exerted in the direction of the line PG, or towards the . centre S, is as the
length PG. Therefore the force of all the points in the plane EF (that is,
of that whole plane) by which the corpuscle P is attracted towards the
centre S is as the distance PG multiplied by the number of those points,
that is, as the solid contained under that plane EF and the distance PG.
And in like manner the force of the plane ef, by which the corpuscle P is
attracted towards the centre S, is as that plane drawn into its distance Pg,
or as the equal plane EF drawn into that distance Pg* ; and the sum of the
forces of both planes as the plane EF drawn into the sum of the distances
PG + P^, that is, as that plane drawn into twice the distance PS of the
centre and the corpuscle ; that is, as twice the plane EF drawn into the dis
tance PS, or as the sum of the equal planes EF + ef drawn into the same
distance. And, by a like reasoning, the forces of all the planes in the
whole sphere, equi-distant on each side from the centre of the sphere, are
as the sum of those planes drawn into the distance PS, that is, as the
whole sphere and the distance PS conjunctly. Q,.E.D.

CASE 2. Let now the corpuscle P attract the sphere AEBF. And, by
the same reasoning, it will appear that the force with which the sphere is
attracted is as the distance PS. Q,.E.D.

CASE 3. Imagine another sphere composed of innumerable corpuscles P :
and because the force with which every corpuscle is attracted is as the dis
tance of the corpuscle from the centre of the first sphere, and as the same
sphere conjunctly, and is therefore the same as if it all proceeded from a
single corpuscle situate in the centre of the sphere, the entire force with
which all the corpuscles in the second sphere are attracted, that is, with
which that whole sphere is attracted, will be the same as if that sphere

SEC. Xll.] OP NATURAL PHILOSOPHY. 225

were attracted by a force issuing from a single corpuscle in the centre of
the first sphere ; and is therefore proportional to the distance between the
centres of the spheres. Q,.E.D.

CASE 4. Let the spheres attract each other mutually, and the force will
be doubled, but the proportion will remain. Q..E.D.

CASE 5. Let the corpuscle p be placed within ^- ^\E

the sphere AEBF ; and because the force of the
plane ef upon the corpuscle is as the solid contain
ed under that plane and the distance jog' ; and the
contrary force of the plane EF as the solid con
tained under that plane and the distance joG ; the ^
force compounded of both will be as the difference **
of the solids, that is, as the sum of the equal planes drawn into half the
difference of the distances ; that is, as that sum drawn into joS, the distance
of the corpuscle from the centre of the sphere. And, by a like reasoning,
the attraction of all the planes EF, ef, throughout the whole sphere, that
is, the attraction of the whole sphere, is conjunctly as the sum of all the
planes, or as the whole sphere, and as joS, the distance of the corpuscle from
the centre of the sphere. Q.E.D.

CASE 6. And if there be composed a new sphere out of innumerable cor
puscles such as jo, situate within the first sphere AEBF, it may be proved,
as before, that the attraction, whether single of one sphere towards the
other, or mutual of both towards each other, will be as the distance joS of
the centres. Q, E.D.

PROPOSITION LXXVIII. THEOREM XXXVIII.

If spheres it* the progress from the centre to the circumference be hoivMtv
dissimilar a->id unequable, but similar on every side round about af all
given distances from the centre ; and the attractive force of evsrt/
point be as the distance of the attracted body ; I say, that the entire
force with which two spheres of this kind attract each other mutitallij
is proportional to the distance between the centres of the spheres.
This is demonstrated from the foregoing Proposition, in the same man
ner as Proposition LXXVI was demonstrated from Proposition LXXY.

COR. Those things that were above demonstrated in Prop. X and LXJV,
of the motion of bodies round the centres of conic sections, take place when
all the attractions are made by the force of sphaerical bodies of the condi
tion above described, and the attracted bodies are spheres of the same kind.

SCHOLIUM.

i have now explained the two principal cases of attractions; to wit,
when the centripetal forces decrease in a duplicate ratio of the distances
•r increase in a simple ratio of the distances, causing the bodies in botli

15

226 THE MATHEMATICAL PRINCIPLES [BoOK 1

cases to revolve in conic sections, and composing sphaerical bodies whose
centripetal forces observe the same law of increase or decrease in the recess
from the centre as the forces of the particles themselves do ; which is verv
remarkable. It would be tedious to run over the other cases, whose con
clusions are less elegant and important, so particularly as I have done
these. I choose rather to comprehend and determine them all by one gen
eral method as follows.

LEMMA XXIX.

ff about the centre S there be described any circle as AEB, and about the
centre P there be. also described two circles EF, ef, cutting the Jirst in
E and e, and the line PS in F and f ; and there be let fall to PS the
perpendiculars ED, ed ; I say, that if the distance of the arcs EF; ef
be supposed to be infinitely diminished, the last ratio of the evanscent
linr Dd to the evanescent line Ff is the same as that of the line PE to
the live PS.
For if the line Pe cut the arc EF in q ; and the right line Ee, which

coincides with the evanescent arc Ee, be produced, and meet the right line
PS in T ; and there be let fall from S to PE the perpendicular SG ; then,
because of the like triangles DTE, </'!>, DES, it will be as Dd to Ee so
))T to TE, or DE to ES : and because the triangles, Ee?, ESG (by Lem.
VIII, and Cor. 3, Lem. VII) are similar, it will be as Ee to eq or F/soES
to SG ; and, ex ceqno, as Dd to Ff so DE to SG ; that is (because of the
similar triangles PDE; PGS), so is PE to PS. Q.E.D.

PROPOSITION LXXIX. THEOREM XXXIX.

Suppose a superficies as EFfe to have its breadth infinitely diminished,
and to be just vanishing ; and that the same superficies by its revolu-
tion round the axis PS describes a spherical concavo-convex solid, to
the several equnJ particle* of which there tend equal centripetal forces ;
I soy, that the force with which thit solid attracts a corpuscle situate
in P is in a ratio compounded of the ratio of the solid DE2 X Ff and
the ratio of the force with which the given particle in the place Ff
would attract the same corpuscle.
For if we consider, first, the force of the spherical superficies FE which

SEC. xn.j

OF NATURAL PHILOSOPHY.

227

is generated by the revolution of the arc FE,
and is cut any where, as in r, by the line</6,
the annular part of the super J'cies generated
by the revolution of the arc rE will be as the
lineola Dd, the radius of the sphere PE re-
mainiag the same; as Archimedes has de
monstrated in his Book of the Sphere and
Cylinder. And the force of this super
ficies exerted in the direction of the lines PE
or Pr situate all round in the conical superficies, will be as this annular
superficies itself; that is as the lineola DC/, or, which is the same, as the
rectangle under the given radius PE of the sphere and the lineola DC/ ; but
that force, exerted in the direction of the line PS tending to the centre S,
will be less in the ratio PI) to PE, and therefore will be as PD X DC/.
Suppose now the line DF to be divided into innumerable little equal par
ticles, each of which call DC/, and then the superficies FE will be divided
into so many equal annuli, whose forces will be as the sum of all the rec
tangles PD X DC/, that is, as |PF2 — - |PD2; and therefore as DE-.
Let now the superficies FE be drawn into the altitude F/; and the force
of the solid EF/e exerted upon the corpuscle P will be as DE2 X Ff;
that is, if the force be given which any given particle as Ff exerts upon
the corpuscle P at the distance PF. But if that force be not given, the
force of the solid EF/e will be as the solid DE2 X Ff and that force not
given, conjunctly. Q.E.D.

PROPOSITION LXXX. THEOREM XL.

If to the several equal parts of a sphere ABE described about the centre
S there tend equal centripetal forces ; and from the several points I)
in the axis of the sphere AB in which a corpuscle, as F, is placed,
there be erected the perpendiculars DE meeting the sphere in E, and
if in those perpendiculars the lengths DN be taken as the quantity

DE2 X PS

— —-, , and as th* force which a particle of the sphere situate in,

the axis exerts at the distance PE upon the corpuscle P conjunctly ; ]
say, that the in hole force with which the, corpuscle P is attracted to
wards the sphere is as the area ANB, comprehended under the axis of
the sphere AB, and the curve line ANB, the locus of the point N.
For supposing the construction in the last Lemma and Theorem to
stand, conceive the axis of the sphere AB to be divided into innumerable
equal particles DC/, and the whole sphere to be divided into so many sphe
rical concavo-convex laminae EF/e / and erect the perpendicular dn. By
the last Theorem, the force with which the laminas EF/e attracts the cor
puscle P is as DE2 X Ff and the force of one particle exerted at the

228

THE MATHEMATICAL PRINCIPLES

[BOOK I.

distance PE or PF, conjunctly.
But (by the last Lemma) Dd is to
F/ as PE to PS, and therefore F/

.
is equal to

PE

F/ is equal to Dd X

; and DE2 X

DE2 X PS
PET~;

and therefore the force of the la-

DE2 X PS
mina EF/e is as Do? X PT?~

and the force of a particle exerted at the distance PF conjunctly ; that is,
by the supposition, as DN X D(/7 or as the evanescent area DNwrf.
Therefore the forces of all the lamina) exerted upon the corpuscle P are as
all the areas DN//G?, that is, the whole force of the sphere will be as the
whole area ANB. Q.E.D.

COR. 1. Hence if the certripetal force tending to the several particles

•p)F2 vx po
remain always the same at all distances, and DN be made as ™ ;

Jr Jli

the whole force with which the corpuscle is attracted by the sphere is as
the area ANB.

COR. 2. If the centripetal force of the particles be reciprocally as the

DE2 X PS

distance of the corpuscle attracted by it, and DN be made as -— ^^ ,

the force with which the corpuscle P is attracted by the whole sphere wil]
be as the area ANB.

Cor. 3. Jf the centripetal force of the particles be reciprocally as the
cube of the distance of the corpuscle attracted by it, and DN be made as

T)F2 y PS

— --- — — . the force with which the corpuscle is attracted by the whole

sphere will be as the area ANB.

COR. 4. And universally if the centripetal force tending to the several
particles of the sphere be supposed to be reciprocally as the quantity V ;

DE2 X PS

and D5& be made as „„ — ^- ; the force with which a corpuscle is at-

Jr Jtj X '

tracted by the whole sphere will be as the area ANB.

PROPOSITION LXXXI. PROBLEM XLI.

T/Le things remaining as above, it is required lo measure the area

ANB.

From the point P let there be drawn the right line PH touching the
sphere in H ; and to the axis PAB, letting fall the perpendicular HI,
bisect PI in L; and (by Prop. XII, Book II, El em.) PE2 is equal tf

SEC. XII.]

OF NATURAL PHILOSOPHY.

229

PS3 + SE2 + 2PSD. But because

the triangles SPH, SHI are alike,

SE2 or SH2 is equal to the rectan

gle PSI, Therefore PE2 is equal

to the rectangle contained under PS

and PS -f SI + 2SD ; that is, under

PS and 2LS + 2SD ; that is, under

PS and 2LD. Moreover DE2 is

equal to SE2 — SD% or SE2 —

LS2 + 2SLD — LD2, that is, 2SLD — LD2 — ALB. For LS°- —

SE2 or LSa — SAa (by Prop. VI, Book II, Elem.) is equal to the rectan

gle ALB. Therefore if instead of DE 2 we write 2SLD — LD 2 — ALB,

the quantity -™ - ^-, which (by Cor. 4 of the foregoing Prop.) is as

PE x

the length of the ordinate DN, will
2SLD x PS LD2 X PS

now

resolve itself into three parts

ALB xPS ...

-TE3rr~ -pfixT" -pE^-v-;whereifinsteadofVwewnt

the inverse ratio of the centripetal force, and instead of PE the mean pro
portional between PS and 2LD, those three parts will become ordinates to
so many curve lines, whose areas are discovered by the common methods.
Q.E.D.

EXAMPLE 1. If the centripetal force tending to the several particles of
the sphere be reciprocally as the distance ; instead of V write PE the dis
tance, then 2PS X LD for PE2 ; and DN will become as SL — £ LD —

•ny |y Suppose DN equal to its double 2SL — LD — - — r^ 5 an<* 2SL

the given part of the ordinate drawn into the length AB will describe the
rectangular area 2SL X AB ; and the indefinite part LD, drawn perpen
dicularly into the same length with a continued motion, in such sort as in
its motion one way or another it may either by increasing or decreasing re-

LB2-LA2

main always equal to the length LD, will describe the area — — ^ — — ,

that is, the area SL X AB ; which taken from the former area 2SL X

AB, leaves the area SL X AE. But the third part - ---, drawn after the

i lit,

same manner with a continued motion perpendicularly into the same length,

will describe the area of an hyperbola, which subducted

from the area SL X AB will leave ANB the area sought.

Whence arises this construction of the Problem. At

the points, L, A, B, erect the perpendiculars L/, Act, B6;

making Aa equal to LB, and Bb equal to LA. Making

L/ and LB asymptotes, describe through the points a, 6,

230

THE MATHEMATICAL PRINCIPLES

[BOOK 1

the hyperbolic crrve ab. And the chord ba being drawn, will inclose the
area aba equal to the area sought ANB.

EXAMPLE 2. If the centripetal force tending to the several particles of
the sphere be reciprocally as the cube of the distance, or (which is the same

PE3

thing; as that cube applied to any given plane ; write
2PS X LD for PE2 ; and DN will become as

'2AS2
SL X AS2

for V, and

AS2

ALB X AS2
2PS X LD2
LSI

PS X LD 2PS

that is (because PS, AS, SI are continually proportional), as
ALB X SI

2LD:
LSI

If we draw then these three parts into th

length AB, the first r-pr will generate the area of an hyperbola ; the sec-

L-t \J

, ALB X SI . ALB X SI

ond iSI the area } AB X SI ; the third 2Ll^ — area - 2LA

, that is, !AB X SI. From the first subduct the sum of the
2LB

second and third, and there will remain ANB, the area sought. Whence
arises this construction of the problem. At the points L, A, S, B, erect
the perpendiculars L/ Aa Ss, Bb, of which suppose Ss
equal to SI ; and through the point s, to the asymptotes
L/, LB, describe the hyperbola asb meeting the
perpendiculars Aa, Bb, in a and b ; and the rectangle
2ASI, subducted from the hyberbolic area AasbB, will
B leave ANB the area sought.

.. ,, . „

EXAMPLE 3. If the centripetal force tending to the several particles of
the spheres decrease in a quadruplicate ratio of the distance from the par-

pT^4 _

tides ; write ~|f- for V, then V 2PS + LD for PE, and DN will become

___
V2SI X

SI2 X ALB

2v2SI

X

These three parts drawn into the length AB, produce so many areas, viz.

J-L

2SI2 X SL . 1

x^ — into

T— r
LA

~~~5ot in*0 V LB — V LA; and
BS12 X ALB . "1 1"

VLA3 v/LB3
And these after due reduction come

forth __

SEC. XII.] OF NATURAL PHILOSOPHY. 2'3\.

2SI3 4 SI3

~oj-p And these by subducting the last from the first, become -oT~r

Therefore the entire force with ,7hich the corpuscle P is attracted towards
the centre of the sphere is as-^, that is, reciprocally as PS3 X PJ

Q.E.I.

By the same method one may determine the attraction of a corpuscle
situate within the sphere, but more expeditiously by the following Theorem.

PROPOSITION LXXXIL THEOREM XLI.

In a sphere described about the centre S with the interval SA, if there be
taken SI, SA, SP continually proportional ; ! sat/, that the attraction,
of a corpuscle within the sphere in any place I is to its attraction without
the sphere in the place P in a ratio compounded of the subduplicate
ratio of IS, PS, the distances from the centre, and the subduplicate
ratio of tJie centripetal forces tending to the centre in those places P
and I.

As if the centripetal forces of the
particles of the sphere be reciprocally
;is the distances of the corpuscle at
tracted by them ; the force with which
the corpuscle situate in I is attracted
by the entire sphere will be to the
force with which it is attracted in P
in a ratio compounded of the subdu
plicate ratio of the distance SI to the distance SP, and the subduplicate
ratio of the centripetal force in the place I arising from any particle in the
centre to the centripetal force in the place P arising from the same particle in
the centre ; that is, in the subduplicate ratio of the distances SI, SP to each
other reciprocally. These two subduplicate ratios compose the ratio of
equality, and therefore the attractions in I and P produced by the whole
sphere are equal. By the like calculation, if the forces of the particles of
the sphere are reciprocally in a duplicate ratio of the distances, it will be
found that the attraction in I is to the attraction in P as the distance SP
to the semi -diameter SA of the sphere. If those forces are reciprocally in
a triplicate ratio of the distances, the attractions in I and P will be to each
other as SP2 to SA3 ; if in a quadruplicate ratio, as SP3 to SA3. There
fore since the attraction in P was found in this last case to be reciprocally
as PS 3 X PI, the attraction in I will be reciprocally as S A 3 X PI, that is,
because S A 3 is given reciprocally as PI. And the progression is the same
in injinitnm. The demonstration of this Theorem is as follows :

The things remaining as above constructed, and a corpuscle being in anj

332

THE MATHEMATICAL PRINCIPLES

[BOOK I.

place P. the ordinate DN was found to be as

T)F2 \" PS

— 00 ^~\r- Therefore if

I Cj X V

IE be drawn, that ordinate for any other place of the corpuscle, as I, will

DE2 X IS

become (mutatis mutandis] as ~T~p~rry~- Suppose the centripetalsorces

flowing from any point of the sphere, as E, to be to each other at the dis
tances IE and PE as PE'1 to IE11 (where the number u denotes the index

DE2 X PS

of the powers of PE and IE), and those ordinates will become as ^p - -57^7,

2 \x IS

and — ~" --- TT7, whose ratio to each other is as PS X IE X IEn to IS X

IE X IE"

PE X PEn. Because SI, SE, SP are in continued proportion, the tri
angles SPE, SEI are alike ; and thence IE is to PE as IS to SE or SA.
For the ratio of IE to PE write the ratio of IS to SA ; and the ratio of
the ordinates becomes that of PS X IE" to SA X PEn. But the ratio of
PS to SA is snbduplicate of that of the distances PS, SI ; and the ratio of
IE" to PE1 (because IE is to PE as IS to SA) is subduplicate of that of
the forces at the distances PS, IS. Therefore the ordinates, and conse
quently the areas whioifi the ordinates describe, and the attractions propor
tional to them, are in a ratio compounded of those subduplicate ratios.
Q.E.D.

PROPOSITION LXXXIII. PROBLEM XLII.

To find the force with which a corpuscle placed in the centre of a sphere
is attracted towards any segment of that sphere whatsoever.

Let P be a body in the centre of that sphere and
RBSD a segment thereof contained under the plane
RDS, and thesphrcrical superficies RBS. Let DB be cut
in F by a sphaerical superficies EFG described from the
centre P, and let the segment be divided into the parts
_B BREFGS, FEDG. Let us suppose that segment to
be not a purely mathematical but a physical superficies,
having some, but a perfectly inconsiderable thickness.
* Let that thickness be called O, and (by what Archi
medes has demonstrated) that superficies will be as
PF X DF X O. Let us suppose besides the attrac
tive forces of the particles of the sphere to be reciprocally as that power of
r.he distances, of which n is index ; and the force with which the superficies

DE2 X O

EFG attracts the body P will be (by Prop. LXXIX) as -- — that,

2DF X O
is, as ---?— -,-

DF2 X O
~"~ppn — *

ppn

the perpendicular FN drawn into

SEC. XJ11.I

OF NATURAL PHILOSOPHY.

233

O be proportional to this quantity ; and the curvilinear area BDI, which
the ordinate FN, drawn through the length DB with a continued motion
will describe, will be as the whole force with which the whole segment
RBSD attracts the body P. Q.E.I.

PROPOSITION LXXXIV. PROBLEM XLIII.

To find the force with which a corpuscle, placed without the centre of a
sphere iti the axis of any segment, is attracted by that segment.
Let the body P placed in. the axis ADB of

the segment KBK be attracted by that seg
ment. About the centre P, with the interval

PE, let the spherical superficies EFK be de-

scribed; and let it divide the segment into

two parts EBKFE and EFKDE. Find the

force of the first of those parts by Prop.

LXXXI, and the force of the latter part by

Prop. LXXXIII, and the sum of the forces will be the force of the whole

segment EBKDE. Q.E.I.

SCHOLIUM.

The attractions of sphaerical bodies being now explained, it comes next
in order to treat of the laws of attraction in other bodies consisting in like
manner of attractive particles ; but to treat of them particularly is not neces
sary to my design. It will be sufficient to subjoin some general proposi
tions relating to the forces of such bodies, and the motions thence arising,
because the knowledge of these will be of some little use in philosophical
inquiries.

SECTION XIII.

Of the attractive forces of bodies which are not of a sphcerical figure.

PROPOSITION LXXXV. THEOREM XLIL

If a body be attracted by another, and its attraction be vastly stronger
when it is contiguous to the attracting body than when they are sepa
rated from one another by a very small interval ; the forces of the
particles of the attracting body decrease, in the recess of the body at
tracted, in more than a duplicate ratio of the distance of the particles.
For if the forces decrease in a duplicate ratio of the distances from the
particles, the attraction towards a sphaerical body being (by Prop. LXXIV)
reciprocally as the square of the distance of the attracted body from the
centre of the sphere, will not be sensibly increased by the contact, and it

234 THE MATHEMATICAL PRINCIPLES [BOOK 1

\vill be still less increased by it, if the attraction, in the recess of the body
attracted, decreases in a still less proportion. The proposition, therefore,
is evident concerning attractive spheres. And the case is the same of con
cave sphaerical orbs attracting external bodies. And much more does it
appear in orbs that attract bodies placed within them, because there the
attractions diffused through the cavities of those orbs are (by Prop. LXX)
destroyed by contrary attractions, and therefore have no effect even in the
place of contact. Now if from these spheres and sphoerical orbs we take
away any parts remote from the place of contact, and add new parts any
where at pleas ore, we may change the figures of the attractive bodies at
pleasure ; but the parts added or taken away, being remote from the place
of contact, will cause no remarkable excess of the attraction arising from
the contact of the two bodies. '1 herefore the proposition holds good in
bodies of all figures. Q.E.I).

PROPOSITION LXXXV1. THEOREM XLIII.

If the forces of the particles of which an attractive body is composed de
crease, in. the recess of the attractive body, in a triplicate or more than
a triplicate ratio of the distance from the particles, the attraction will
be vastly stronger in the point of contact than when the attracting and
attracted bodies are separated from each other, though by never so
small an interval.

For that the attraction is infinitely increased when the attracted corpus
cle comes to touch an attracting sphere of this kind, appears, by the solu
tion of Problem XLI, exhibited in ££e second and third Examples. The
same will also appear (by comparing those Examples and Theorem XLI
together) of attractions of bodies made towards concavo-convex orbs, whether
the attracted bodies be placed without the orbs, or in the cavities within
them. And by aiding to or taking from those spheres and orbs any at
tractive matter any where without the place of contact, so that the attrac
tive bodies may receive any assigned figure, the Proposition will hold good
of all bodies universally. Q.E.D.

PROPOSITION LXXXVII. THEOREM XI. IV.

If two bodies similar to each other, and consisting of matter equally at
tractive attract separately two corpuscles proportional to those bodies,
and in a like situation to them, the accelerative attractions of the cor
puscles towards the entire bodies will be as the acccleratire at tractions
of the corpuscles towards particles of the bodies proportional to the
wholes, and alike situated in them.

For if the bodies are divided into particles proportional to the wholes,
and alike situated in them, it will be, as the attraction towards any parti
cle of one of the bodies to the attraction towards the correspondent particle

SEC. A III.] OF NATURAL PHILOSOPHY. 235

in the other body, so are the attractions towards the several particles of the
iirst body, to the attractions towards the several correspondent particles of
the other body ; and, by composition, so is the attraction towards the first
whole body to the attraction towards the second whole body. Q,.E.U.

COR. 1 . Therefore if, as the distances of the corpuscles attracted increase,
the attractive forces of the particles decrease in the ratio of any power
of the distances, the accelerative attractions towards the whole bodies will
be as the bodies directly, and those powers of the distances inversely. A*
if the forces of the particles decrease in a duplicate ratio of the distances
from the corpuscles attracted, and the bodies are as A 3 and B 3, and there
fore both the cubic sides of the bodies, and the distance of the attracted
corpuscles from the bodies, are as A and B ; the accelerative attractions

A3 B3

towards the bodies will be as — and — , that is, as A and B the cubic

sjides of those bodies. If the forces of the particles decrease in a triplicate
ratio of the distances from the attracted corpuscles, the accelerative attrac-

A3 B3

tions towards the whole bodies will be as — and 5--, that is, equal. If the

A. tj

forces decrease in a quadruplicate ratio, the attractions towards the bodies

A3 B3

will be as-— an^ 04' *^at is, reciprocally as the cubic sides A and B.

And so in other cases.

COR. 2. Hence, on the other hand, from the forces with which like bodies
attract corpuscles similarly situated, may be collected the ratio of the de
crease of the attractive forces of the particles as the attracted corpuscle
recedes from them ; if so be that decrease is directly or inversely in any
ratio of the distances.

PROPOSITION LXXXVIII. THEOREM XLV.

If the attractive forces of the equal particles of any body be as the dis
tance of the places from the particles, the force of the whole body will
tend to its centre of gravity ; and will be the same with the force of
a globe, consisting of similar and equal matter, and having its centre
in the centre of gravity.
Let the particles A, B, of the body RSTV at
tract any corpuscle Z with forces which, suppos-|
ing the particles to be equal between themselves,
are as the distances AZ, BZ ; but, if they are
supposed unequal, are as those particles and
their distances AZ, BZ, conjunctly, or (if I may
go speak) as those particles drawn into their dis
tances AZ, BZ respectively. And let those forces be expressed by the

236 THE MATHEMATICAL PRINCIPLES [BOOK 1.

contents u.ider A X AZ, and B X BZ. Join AB, and let it be cut in G,
so that AG may be to BG as the particle B to the particle A : and G
will be the common centre of gravity of the particles A and B. The force
A X AZ will (by Cor. 2, of the Laws) be resolved into the forces A X GZ
and A X AG ; and the force B X BZ into the forces B X GZ and B X
BG. Now the forces A X AG and B X BG, because A is proportional to
B, and BG to AG, are equal, and therefore having contrary directions de
stroy one another. There remain then the forces A X GZ and B X GZ.
These tend from Z towards the centre G, and compose the force A + B
X GZ ; that is, the same force as if the attractive particles A and B were
placed in their common centre of gravity G, composing there a little globe.

By the same reasoning, if there be added a third particle G, and the
force of it be compounded with the force A -f B X GZ tending to the cen
tre G, the force thence arising will tend to the common centre of gravity
of that globe in G and of the particle C ; that is, to the common centre oi
gravity of the three particles A, B, C ; and will be the same as if that
globe and the particle C were placed in that common centre composing a
greater globe there ; and so we may go on in injinitum. Therefore
the whole force of all the particles of any body whatever RSTV is the
same as if that body, without removing its centre of gravity, were to put
on the form of a globe. Q,.E.D.

COR. Hence the motion of the attracted body Z will be the same as if
the attracting body RSTV were sphaerical ; and therefore if that attract
ing body be either at rest, or proceed uniformly in a right line, the body
attracted will move in an ellipsis having its centre in the centre of gravity
of the attracting body.

PROPOSITION LXXXIX. THEOREM XLVI.

If there be several bodies consisting of equal particles whose jorces are
as the distances of the places from each, the force compounded of all
the forces by which any corpuscle is attracted will tend to the common
centre of gravity of the attracting bodies ; and will be the same as if
those attracting bodies, preserving their common centre of gravity,
should unite there, and be formed into a globe.

This is demonstrated after the same manner as the foregoing Proposi
tion.

COR. Therefore the motion of the attracted body will be the same as if
the attracting bodies, preserving their common centre of gravity, should
unite there, and be formed into a globe. And, therefore, if the common
centre of gravity of the attracting bodies be either at rest, or proceed uni
formly in a right line, the attracted body will move in an ellipsis having
Us centre in the common centre of gravity of the attracting bodies.

SEC. XlII.j OF NATURAL PHILOSOPHY. 237

PROPOSITION XC. PROBLEM XLIV.

If to the several points of any circle there tend equal centripeta forces,
increasing' or decreasing in any ratio of the distances ; it is required
to Jin d the force with which a corpuscle is attracted, that is, situate
any where in a right line which stands at right angles to the plant
of the circle at its centre.
Suppose a circle to be described about the cen
tre A with any interval AD in a plane to which ;
the right line AP is perpendicular ; and let it be
required to find the force with which a corpuscle
P is attracted towards the same. From any point
E of the circle, to the attracted corpuscle P, let
there be drawn the right line PE. In the right
line PA take PF equal to PE, and make a per-
pendicular FK, erected at F, to be as the force
with which the point E attracts the corpuscle P.
And let the curve line IKL be the locus of the point K. Let that cu/, fe
meet the plane of the circle in L. In PA take PH equal to PD, and p/^ct
the perpendicular HI meeting that curve in I ; and the attraction of the
corpuscle P towards the circle will be as the area AHIL drawn into the
altitude AP Q.E.I.

For let there be taken in AE a very small line Ee. Join Pe, and in PE,
PA take PC, Pf equal to Pe. And because the force, with which any
point E of the annulus described about the centre A with the interval AS
in the aforesaid plane attracts to itself the body P, is supposed to be as
FK ; and, therefore, the force with which that point attracts the body P

AP X FK

towards A is as - — ^p ; and the force with which the whole annulus

AP X FK

attracts tne body P towards A is as the annulus and p^ conjunct-

ly ; and that annulus also is as the rectangle under the radius AE aad the
breadth Ee, and this rectangle (because PE and AE, Ee and CE are pro
portional) is equal to the rectangle PE X CE or PE X F/; the force
*-ith which that annulus attracts the body P towards A will be as PE X

AP X FK
Ff and pp~~~ conjunctly ; that is, as the content under F/ X FK X

AP, or as the area FKkf drawn into AP. And therefore the sum of the
forces with which all the annuli, in the circle described about the centre A
with the interval AD, attract the body P towards A, is as the whole area
AHIKL drawn into AP. Q.E.D.
COR. 1. Hence if the forces of the points decrease in the duplicate ratio

238

THE MATHEMATICAL PRINCIPLES

[BOOK I

of the distances, that is, if FK be as rfFK, and therefore the area AHIKL

as p-7 — p- ; the attraction of the corpuscle P towards the circle will

PA AH

be as 1 — ; that is, as

COR. 2. And universally if the forces of the points at the distances D b(
reciprocally as any power Dn of the distances; that is, if FK be as — .

and therefore the area AHIKL as

1

1

" — l PH" —

1 PA

, ; the attraction

of the corpuscle P towards the circle will be as

PA" — 2 PH" — l

COR. 3. And if the diameter of the circle be increased in itifinitum, and
the number n be greater than unity ; the attraction of the corpuscle P to
wards the whole infinite plane will be reciprocally as PA" — 2, because the

PA

other term vanishes.

PROPOSITION XCI. PROBLEM XLV.

To find the attraction of a corpuscle situate in the axis of a round solid,
to whose several points there tend equal centripetal forces decreasing
in any ratio of the distances whatsoever.

Let the corpuscle P, situate in the axis AB
of the solid DECG, be attracted towards that
solid. Let the solid be cut by any circle as
RFS, perpendicular to the axis ; and in its
semi-diameter FS, in any plane PALKB pass
ing through the axis, let there be taken (by
Prop. XC) the length FK proportional to the
force with which the corpuscle P is attracted
towards that circle. Let the locus of the point
K be the curve line LKI, meeting the planes of the outermost circles AL
and BI in L and I ; and the attraction of the corpuscle P towards the
solid will be as the area LABI. Q..E.I.

COR. 1. Hence if the solid be a cylinder described by the parallelogram
A DEB revolved about the axis AB, and the centripetal forces tending to
the several points be reciprocally as the squares of the distances from the
points ; the attraction of the corpuscle P towards this cylinder will be as
AB — PE + PD. For the ordinate FK (by Cor. 1, Prop. XC) will be

PF

as 1 — --. The part 1 of this quantity, drawn into the length AB, de-

SEC. XIII.

OF NATURAL PHILOSOPHY

239

scribes the area 1 X AB ; and the other part

PF

, drawn into the length PB describes the

ix

area 1 into PE — AD (as may be easily
shewn from the quadrature of the curve
LKI); and, in like manner, the same part
drawn into the length PA describes the area

L into PD — AD. and drawn into AB, the

"At

G

Iv

S

13

M

7J"

1

difference of PB and PA, describes 1 into PE — PD, the difference of the
areas. From the first content 1 X AB take away the last content 1 into
PE — PD, and there will remain the area LABI equal to 1 into
AB — PE -h PD. Therefore the force, being proportional to this area,
is as AB — PE + PD.

COR. 2. Hence also is known the force
by which a spheroid AGBC attracts any
body P situate externally in its axis AB.
Let NKRM be a conic section whose or-
dinate KR perpendicular to PE may be \
always equal to the length of the line PD,
continually drawn to tlie point D in
which that ordinate cuts the spheroid.
From the vertices A, B, of the spheriod,
let there be erected to its axis AB the perpendiculars AK, BM, respectively
equal to AP. BP, and therefore meeting the conic section in K and M; and
join KM cutting off from it the segment KMRK. Let S be the centre of the
spheroid, and SC its greatest semi-diameter : and the force with which the
spheroid attracts the body P will be to the force with which a sphere describ-

, ....,,. ASxCS2-PSxKMRK

ed with the diameter AhJ attracts the same body as prrr ^ r-=

1 o -f- Go2 — Ao

AS3
is to fkT^,. And by a calculation founded on the same principles may be

found the forces of the segments of the spheroid.

COR. 3. If the corpuscle be placed within the spheroid and in its axis,
the attraction will be as its distance from the centre. This may be easily
collected from the following reasoning, whether
the particle be in the axis or in any other given
diameter. Let AGOF be an attracting sphe
roid, S its centre, and P the body attracted.
Through the body P let there be drawn the
semi-diameter SPA, and two right lines DE,
FC meeting the spheroid in 1) and E, F and
G ; and let PCM, HLN be the superficies of

240 THE MATHEMATICAL PRINCIPLE* ffioOK 1.

two interior spheroids similar and concentrical to the exterior, the first of
which passes through the body P. and cuts the right lines DE, FG in B
and C ; arid the latter cuts the same right lines in H and I, K and L.
I ,et the spheroids have all one common axis, and the parts of the right
lines intercepted on both sides DP and BE, FP and CG, DH and IE, FK
and LG, will be mutually equal; because the right lines DE. PB, and HI.
are bisected in the same point, as are also the right lines FG, PC, and KL.
Conceive now DPF. EPG to represent opposite cones described with the
infmitely small vertical angles DPF, EPG, and the lines DH, El to be
infinitely small also. Then the particles of the cones DHKF, GLIE, cut
off by the spheroidical superficies, by reason of the equality of the lines DH
and ET; will be to one another as the squares of the distances from the body
P, and will therefore attract that corpuscle equally. And by a like rea
soning if the spaces DPF, EGCB be divided into particles by the superfi
cies of innumerable similar spheroids concentric to the former and having

J. O

one common axis, all these particles will equally attract on both sides the
body P towards contrary parts. Therefore the forces of the cone DPF.
and of the conic segment EGCB, are equal, and by their contrariety de
stroy each other. And the case is the same of the forces of all the matter
that lies without the interior spheroid PCBM. Therefore the body P is
attracted by the interior spheroid PCBM alone, and therefore (by Cor. 3,
Prop. 1 , XXII) its attraction is to the force with which the body A is at
tracted by the whole spheroid AGOD as the distance PS to the distance
AS. Q.E.D.

PROPOSITION XCII. PROBLEM XLVI.

An attracting body being given, it is required to find the ratio of the de
crease of the centripetal forces tending to its several points.
The body given must be formed into a sphere, a cylinder, or some regu
lar figure, whose law of attraction answering to any ratio of decrease may
be found by Prop. LXXX, LXXXI, and XCI. Then, by experiments,
the force of the attractions must be found at several distances, and the law
of attraction towards the whole, made known by that means, will give
the ratio of the decrease of the forces of the several parts ; which was to
be found.

PROPOSITION XCIII. THEOREM XLVII.

If a solid be plane on one side, and infinitely extended on all otljer sides,
and consist of equal particles equally attractive, iv hose forces decrease,
in the recess from the solid, in the ratio of any power greater than the
square of the distances ; and a corpuscle placed towards eit far part of
the plane is attracted by the force of the whole solid ; I say that tfie
attractive force of the whole solid, in the recess from its platw superfi-

XIILj

OF NATURAL PHILOSOPHY".

241

n

H

m

G

ties, will decrease in the ratio of a power whose side is the distance oj

the corpuscle from the plane, and its index less by 3 than the index oj

the power of the distances.

CASE 1. Let LG/be the plane by which
the solid is terminated. Let the solid
lie on that hand of the plane that is to
wards I, and let it be resolved into in- _.
numerable planes mHM, //IN, oKO,
(fee., parallel to GL. And first let the
attracted body C be placed without the
solid. Let there be drawn CGHI per
pendicular to those innumerable planes,
and let the attractive forces of the points of the solid decrease in the ratio
of a power of the distances whose index is the number n not less than 3.
Therefore (by Cor. 3, Prop. XC) the force with which any plane mHM
attracts the point C is reciprocally as CHn— 2. In the plane mHM take the
length HM reciprocally proportional to CH1— 2, and that force will be as
HM. In like manner in the several planes /GL, //,TN, oKO, (fee., take the
lengths GL, IN, KO, (fee., reciprocally proportional to CGn— 2, CI1— 2,
CKn — 2, (fee., and the forces of those planes will bs as the lengths so taken,
and therefore the sum of the forces as the sum of the lengths, that is, the
force of the Avhole solid as the area GLOK produced infinitely towards
OK. But that area (by the known methods of quadratures) is reciprocally
as CGn— 3, and therefore the force of the whole solid is reciprocally as
CG"-3. Q.E.D.

CASE 2. Let ttecorpuscleC be now placed on that
hand of the plane /GL that is within the solid,
and take the distance CK equal to the distance
CG. And the part of the solid LG/oKO termi
nated by the parallel planes /GL, oKO, will at
tract the corpuscle C, situate in the middle, neither
one way nor another, the contrary actions of the
opposite points destroying one another by reason of
their equality. Therefore the corpuscle C is attracted by the force only
of the solid situate beyond the plane OK. But this force (by Case 1) is
reciprocally as CKn—3, that is, (because CG, CK are equal) reciprocally as
CG"— 3. Q,.E.D.

COR. 1. Hence if the solid LGIN be terminated on each sitfe by two in
finite parallel places LG, IN, its attractive force is known, subducting
from the attractive force of the whole infinite solid LGKO the attractive
force of the more distant part NIKO infinitely produced towards KO.

COR. 2. If the more distant part of this solid be rejected, because its at
traction compared with the attraction of the nearer part is inconsiderable,

16

0 N

K 1
0

C

242 THE MATHEMATICAL PRINCIPLES [BOOK 1

the attraction of that nearer part will, as the distance increases, decrease
nearly in the ratio of the power CGn— 3.

Con. 3. And hence if any finite body, plane on one side, attract a cor
puscle situate over against the middle of that plane, and the distance between
the corpuscle and the plane compared with the dimensions of the attracting
body be extremely small ; and the attracting body consist of homogeneous
particles, whose attractive forces decrease in the ratio of any power of the
distances greater than the quadruplicate ; the attractive force of the whole
body will decrease very nearly in the ratio of a power whose side is that
very small distance, and the index less by 3 than the index of the former
power. This assertion does not hold good, however, of a body consisting
of particles whose attractive forces decrease in the ratio of the triplicate
power of the distances ; because, in that case, the attraction of the remoter
part of the infinite body in the second Corollary is always infinitely greater
than the attraction of the nearer part.

SCHOLIUM.

If a body is attracted perpendicularly towards a given plane, and from
the law of attraction given, the motion of the body be required ; the Pro
blem will be solved by seeking (by Prop. XXXIX) the motion of the body
descending in a right line towards that plane, and (by Cor. 2, of the Laws)
compounding that motion with an uniform motion performed in the direc
tion of lines parallel to that plane. And, on the contrary, if there be re
quired the law of the attraction tending towards the plane in perpendicu
lar directions, by which the body may be caused to move in any given
curve line, the Problem will be solved by working after the manner of the
third Problem.

But the operations may be contracted by resolving the ordinates into
converging series. As if to a base A the length B be ordinately ap
plied in any given angle, and that length be as any power of the base

A^ ; and there be sought the force with which a body, either attracted to
wards the base or driven from it in the direction of that ordinate, may be
caused to move in the curre line which that ordinate always describes with
its superior extremity ; I suppose the base to be increased by a very small

,m m

part O, and I resolve the ordinate A -f Ol^ into an infinite series A- -f

!!L OA ^ + '^-^--- OOA — ;- &c., and I suppose the force proper-
11 £1111

tional to the term of this series in which O is of two dimensions, that is,
to the term - - OOA ^YT, Therefore the force sought is aa

SEC. XIV.J OF NATURAL PHILOSOPHY. 2M

mm — mn m — 2n .... mm — mn m — 2n

A — 7, — , or, which is the same thinor, as L> m .

nn nn

As if the ordinate describe a parabola, m being — 2, and n = 1, the force
will be as the given quantity 2B°, and therefore is given. Therefore with
a given force the body will move in a parabola, as Galileo has demon
strated. If the ordinate describe an hyperbola, m being = 0 — 1, and n
— 1, the force will be as 2 A 3 or 2B 3 ; and therefore a force which is as the
cube of the ordinate will cause the body to move in an hyperbola. But
leaving this kind of propositions, I shall go on to some others relating to
motion which I have fiot yet touched upon.

SECTION XIV.

Of the motion of very small bodies when agitated by centripetal forces
tending to the several parts of any very great body.

PROPOSITION XCIV. THEOREM XLVIII.

If two similar mediums be separated from each other by a space termi
nated on both sides by parallel planes, and a body in its passage
through that space be attracted or impelled perpendicularly towards
either of those mediums, and not agitated or hindered by any other
force ; and the attraction be every where the same at equal distances
from either plane, taken towards the same hand of the plane ; I say,
that the sine of incidence upon either plane will be to the sine of emcr
gence from the other plane in a given ratio.
CASE 1. Let Aa and B6 be two parallel planes,
and let the body light upon the first plane Aa in
the direction of the line GH, and in its whole
passage through the intermediate space let it be
attracted or impelled towards the medium of in
cidence, and by that action let it be made to de
scribe a curve line HI, and let it emerge in the di
rection of the line IK. Let there be erected IM
perpendicular to Eb the plane of emergence, and
meeting the line of incidence GH prolonged in M, and the plane of inci
dence Aa in R ; and let the line of emergence KI be produced and meet
HM in L. About the centre L, with the interval LI, let a circle be de
scribed cutting both HM in P and Q, and MI produced in N ; and, first,
if the attraction or impulse be supposed uniform, the curve HI (by what
Galileo has demonstrated) be a parabola, whose property is that of a roc-

THE MATHEMATICAL PRINCIPLES [BoOK 1

tangle under its given latiis rectum and the line IM is equal to the squarrf
cf HM ; and moreover the line HM will be bisected in L. Whence if to
MI there be let fall the perpendicular LO, MO, OR will be equal; and
adding the equal lines ON, OI, the wholes MN, IR will be equal also.
Therefore since IR is given, MN is also given, and the rectangle NMI is
to the rectangle under the latus rectum and IM, that is, to HMa in a given
ratio. But the rectangle NMI is equal to the rectangle PMQ,, that is, to
the difference of the squares ML2, and PL2 or LI2 ; and HM2 hath a given
ratio to its fourth part ML2; therefore the ratio of ML2 — LI2 to ML2 is given,
and by conversion the ratio of LI2 to ML ', and its subduplicate, theratrio
of LI to ML. But in every triangle, as LMI, the sines jf the angles are
proportional to the opposite sides. Therefore the ratio of the sine of the
angle of incidence LMR to the sine of the angle of emergence LIR is
given. QJE.lr).

CASE 2. Let now the body pass successively through several spaces ter
minated with parallel planes Aa/>B, B6cC, &c., and let it be acted on by a
\ . force which is uniform in each of them separ-

\ a ately, but different in the different spaces ; and

B \ fr by what was just demonstrated, the sine of the

c ^^ c angle of incidence on the first plane Aa is to

the sine of emergence from the second plane Bb

in a given ratio ; and this sine of incidence upon the second plane Bb will
be to the sine of emergence from the third plane Cc in a given ratio ; and
this sine to the sine of emergence from the fourth plane Dd in a given ra
tio ; and so on in infinitum ; and, by equality, the sine of incidence on
the first plane to the sine of emergence from the last plane in a given ratio.
I ,et now the intervals of the planes be diminished, and their number be in
finitely increased, so that the action of attraction or impulse, exerted accord
ing to any assigned law, may become continual, and the ratio of the sine of
incidence on the first plane to the sine of emergence from the last plane
being all along given, will be given then also. QJE.D.

PROPOSITION XCV. THEOREM XLIX.

The same thing's being supposed, I say, that the velocity of the body be
fore its incidence is to its velocity after emergence as the sine of emer
gence to the sine of incid nee.

Make AH and Id equal, and erect the perpen-
diculars AG, dK meeting the lines of incidence
and emergence GH, IK, in G and K. In GH
--« take TH equal to IK, and to the plane Aa let
^ fall a perpendicular TV. And (by Cor. 2 of the
|x^ I Laws of Motion) let the motion of the body be
jv- resolved into two, one perpendicular to the planes

SEC. X1V.J OF NATURAL PHILOSOPHY. 245

Aa, Bb, Cc, &c, and another parallel to them. The force of attraction or
impulse, acting in directions perpendicular to those planes, does not at all
alter the motion in parallel directions ; and therefore the body proceeding
with this motion will in equal times go through those equal parallel inter
vals that lie between the line AG and the point H, and between the point
I and the line dK ; that is, they will describe the lines GH, IK in equal
times. Therefore the velocity before incidence is to the velocity after
emergence as GH to IK or TH, that is, as AH or Id to vH, that is (sup
posing TH or IK radius), as the sine of emergence to the sine of inci
dence. Q.E.D.

PROPOSITION XCVL THEOREM L.

The same things being supposed, and that the motion before incidence is
swifter than afterwards ; 1 sat/, lhat if the line of incidence be in
clined continually, the body will be at last reflected, and the angle of
reflexion will be equal to the angle of incidence.

For conceive the body passing between the parallel planes Aa, Bb, Cc,
&c., to describe parabolic arcs as above;
and let those arcs be HP, PQ, QR, &c.
And let the obliquity of the line of inci- g
dence GH to the first plane Aa be such rc~ £

that the sine of incidence may be to the radius of the circle whose sine it is,
in the same ratio which the same sine of incidence hath to the sine of emer
gence from the plane Dd into the space DefeE ; and because the sine of
emergence is now become equal to radius, the angle of emergence will be a
right one, and therefore the line of emergence will coincide with the plane
Dd. Let the body come to this plane in the point R ; and because the
line of emergence coincides with that plane, it is manifest that the body can
proceed no farther towards the plane Ee. But neither can it proceed in the
line of emergence Rd; because it is perpetually attracted or impelled towards
the medium of incidence. It will return, therefore, between the planes Cc,
Dd, describing an arc of a parabola Q,R</, whose principal vertex (by what
Galileo has demonstrated) is in R, cutting the plane Or in the same angle
at q, that it did before at Q, ; then going on in the parabolic arcs qp, ph,
&c., similar and equal to the former arcs QP, PH, &c., it will cut the rest
of the planes in the same angles at p, h, (fee., as it did before in P, H, (fee.,
and will emerge at last with the same obliquity at h with which it first
impinged on that plane at H. Conceive now the intervals of the planes
Aa, Bb, Cc, Dd, Ee, (fee., to be infinitely diminished, and the number in
finitely increased, so that the action of attraction or impulse, exerted ac
cording to any assigned law, may become continual; and, the angle of
emergence remaining all alor g equal to the angle of incidence, will be
equal to the same also at last. Q.E.D.

246 THE MATHEMATICAL PRINCIPLES IBoOK 1

SCHOLIUM.

These attractions bear a great resemblance to the reflexions and refrac
tions of light made in a given ratio of the secants, as was discovered h}
Siiellius ; and consequently in a given ratio of the sines, as was exhibited
by Hes Cortes. For it is now certain from the phenomena of Jupiter's
^satellites, confirmed by the observations of different astronomers, that light
is propagated in succession, and requires about seven or eight minutes to
travel from the sun to the earth. Moreover, the rays of light that are in
our air (as lately was discovered by Grimaldus, by the admission of light
into a dark room through a small hole, which 1 have also tried) in their
passage near the angles of bodies, whether transparent or opaque (such aa
the circular and rectangular edges of gold, silver and brass coins, or of
knives, or broken pieces of stone or glass), are bent or inflected round those
bodies as if they were attracted to them ; and those rays which in their
passage come nearest to the bodies are the most inflected, as if they were
most attracted : which tiling I myself have also carefully observed. And
those which pass at greater distances are less inflected ; and those at still
greater distances are a little inflected the contrary way, and form three
fringes of colours. In the figure 5 represents the edge of a knife, or any

tl \

/

K", \

/

^•::^

x -^» . v

^--:<*,

-f:::r;^c :.-/

N «>V J

V U

W~"~a~" "~a C: O la

kind of wedge AsB : and gowog,fmnif,emtme, dlsld, are rays inflected to
wards the knife in the arcs owo, nvn, mtm, Isl ; which inflection is greater
or less according; to their distance from the knife. Now since this inflec
tion of the rays is performed in the air without the knife, it follows that the
rays which fall upon the knife are first inflected in the air before they touch
the knife. And the case is the same of the rays falling upon glass. The
refraction, therefore, is made not in the point of incidence, but gradually, by
a continual inflection of the rays ; which is done partly in the air before they
touch the glass, partly (if [ mistake not) within the glass, after they have
entered it ; as is represented in the rays ckzc, bujb^ ahxa, falling upon r,
q, p, and inflected between k and z, i and y, h and x. Therefore because
of the analogy there is between the propagation of the rays f light and the
motion of bodies, I thought it not amiss to add the followi »g Propositions
far optical uses ; not at all. considering the nature of the rays of .light, or
inquiring whether they are bodies or not ; but only determining the tra
jectories of bodies which are extremely like the trajectories of the rays.

SEC. XIV.] OF NATURAL PHILOSOPHY. 247

PROPOSITION XCVII. PROBT.-EM XL VII.
Supposing t/w sine of incidence upon any superficies to be in a given ra

tio to the sine of emergence ; and that tha inflection of t/ts paths of

those bodies near that superficies is performed in a very short space,

which may be considered as a point ; it is required to determine suck

a superficies as may cause all the corpuscles issuing from any one

given place to converge to another given place.

Let A be the place from whence the cor- E

puscles diverge ; B the place to which they

should converge ; CDE the curve line which

by its revolution round the axis AB describes . /C

the superficies sought ; D, E, any two points of that curve ; and EF, EG,

perpendiculars let fall on the paths of the bodies AD, DB. Let the point

D approach to and coalesce with the point E ; and the ultimate ratio of

the line DF by which AD is increased, to the line DG by which DB is

diminished, will be the same as that of the sine of incidence to the sine of

emergence Therefore the ratio of the increment of the line AD to the

decrement of the line DB is given: and therefore if in the axis AB there

be taken any where the point C through which the curve CDE must

pass, and CM the increment of AC be taken in that given ratio to CN

the decrement of BC, and from the centres A, B, with the intervals AM,

BN, there be described two circles cutting each other in D ; that point D

will touch the curve sought CDE, and, by touching it any where at pleasure,

will determine that curve. Q.E.I.

COR. 1. By causing the point A or B to go off sometimes in infinitum,
and sometimes to move towards other parts of the point C, will bo obtain
ed all those figures which Cartesins has exhibited in his Optics and Geom
etry relating to refractions. The invention of which Cartcsius having
thought fit to conceal, is here laid open in this Proposition.

COR. 2. If a body lighting on any superfi
cies CD in the direction of a ri^ht line AD, Qj-

O \ »

drawn according to any law, should emerge
in the direction of another right line DK ;
and from the point C there be drawn curve

lines CP, CQ, always perpendicular to AD, DK ; the increments of the
lines PD, QD, and therefore the lines themselves PD, Q.D, generated by
those increments, will be as the sines of incidence and emergence to each
other, and e contra.

PROPOSITION XCVIII. PROBLEM XLVIII.

The same things supposed ; if round the axis AB any attractive super
ficies be described as CD, regular or irregular, through which the bo
dies issuing from the given place A must pass ; it is required to find

THE MATHEMATICAL PRINCIPLES. [BOOK J

a second attractive superficies EF, which may make those bodies con
verge to a given place B.

Let a line joining AB cut
the first superficies in C and
the second in E, the point D
being taken any how at plea
sure. And supposing the
f sine of incidence on the first
superficies to the sine of
emergence from the same, and the sine of emergence from the second super
ficies to the sine of incidence on the same, to be as any given quantity M
to another given quantity N; then produce AB to G, so that BG may be
to CE as M — N to N ; and AD to H, so that AH may be equal to AG ;
arid DF to K, so that DK may be to DH as N to M. Join KB, and about
the centre D with the interval DH describe a circle meeting KB produced
in L, and draw BF parallel to DL; and the point F will touch the line
EF, which, being turned round the axis AB, will describe the superficies
sought. Q.E.F.

For conceive the lines CP, CQ to be every where perpendicular to AD,
DF, and the lines ER, ES to FB, FD respectively, and therefore QS to
be always equal to CK; and (by Cor. 2, Prop. XCVII) PD will be to QD
as M to N, and therefore as DL to DK, or FB to FK ; and by division as
DL — FB or PH — PD — FB to FD or FQ — QD ; arid by composition
as PH— FB to FQ, that is (because PH and CG, QS and CE, are equal),
as CE + BG — FR to CE — FS. But (because BG is to CE as M —
N to N) it. comes to pass also that CE + BG is to CE as M to N; and
therefore, by division, FR is to FS as M to N ; and therefore (by Cor. 2,
Prop XCVI1) the superficies EF compels a body, falling upon it in the
direction DF, to go on in the line FR to the place B. Q.E.D.

SCHOLIUM.

,In the same manner one may go on to three or more superficies. But
of all figures the sphserical is the most proper for optical uses. If the ob
ject glasses of telescopes were made of two glasses of a sphaerical figure,
containing water between them, it is not unlikely that the errors of the
refractions made in the extreme parts of the superficies of the glasses may
be accurately enough corrected by the refractions of the water. Such ob
ject glasses are to be preferred before elliptic and hyperbolic glasses, not only
because they may be formed with more ease and accuracy, but because the
pencils of rays situate without the axis of the glass would be more accu
rately refracted by them. But the different refrangibility of different raya
is the real obstacle that hinders optics from being made perfect by sphaeri
cal or any other figures. Unless the errors thence arising can be corrected,
all the labour spent in correcting the others is quite thrown away.

BOOK II

BOOK II.

OF THE MOTION OF BODIES.

SECTION I.
Of the motion of bodies that are resisted in the ratio of the velocity.

PROPOSITION I. THEOREM I.

Tf a body is resisted in the ratio of its velocity, the motion lost by re
sistance is as the space gone over in its motion.

For since the motion lost in each equal particle of time is as the velocity,
that is, as the particle of space gone over, then, by composition, the motion
lost in the whole time will he as the whole space gone over. Q.E.D.

COR. Therefore if the body, destitute of all gravity, move by its innate
force only in free spaces, and there be given both its whole motion at the
beginning, and also the motion remaining after some part of the way is
gone over, there will be given also the whole space which the body can de
scribe in an infinite time. For that space will be to the space now de
scribed as the whole motion at the beginning is to the part lost of that
motion.

LEMMA I.

Quantities proportional to their differences are continually proportional.
Let A be to A — B as B to B — C and C to C — D, (fee., and, by con
version, A will be to B as B to C and C to D, &c. Q.E.D.

PROPOSITION II. THEOREM II.

If a body is resisted in the ratio of its velocity, and moves, by its vis in-
sita only, through a similar medium, and the times be taken equal,
the velocities in the beginning of each of the times are in a geometri
cal progression, and the spaces described in each of the times are as
the velocities.

CASE 1. Let the time be divided into equal particles ; and if at the very
beginning of each particle we suppose the resistance to act with one single
impulse which is as the velocity, the decrement of the velocity in each of

THE MATHEMATICAL PRINCIPLES [BOOK II.

the particles of time will be as the same velocity. Therefore the veloci
ties are proportional to their differences, and therefore (by Lem. 1, Book
II) continually proportional. Therefore if out of an equal number of par
ticles there be compounded any equal portions of time, the velocities at the
beginning of those times will be as terms in a continued progression, which
are taken by intervals, omitting every where an equal number of interme
diate terms. But the ratios of these terms are compounded of the equaj
ratios of the intermediate terms equally repeated, and therefore are equal
Therefore the velocities, being proportional to those terms, are in geomet
rical progression. Let those equal particles of time be diminished, and
their number increased in infinitum, so that the impulse of resistance may
become continual ; and the velocities at the beginnings of equal times, al
ways continually proportional, will be also in this case continually pro
portional. Q.E.D.

CASE 2. And, by division, the differences of the velocities, that is, the
parts of the velocities lost in each of the times, are as the wholes ; but the
spaces described in each of the times are as the lost parts of the velocities
(by Prop. 1, Book I), and therefore are also as the wholes. Q.E.D.
TT COROL. Hence if to the rectangular asymptotes AC, CH,

the hyperbola BG is described, and AB, DG be drawn per
pendicular to the asymptote AC, and both the velocity of
. the body, and the resistance of the medium, at the very be
ginning of the motion, be expressed by any given line AC,
and, after some time is elapsed, by the indefinite line DC ; the time may
be expressed by the area ABGD, and the space described in that time by
the line AD. For if that area, by the motion of the point D, be uniform
ly increased in the same manner as the time, the right line DC will de
crease in a geometrical ratio in the same manner as the velocity ; and the
parts of the right line AC, described in equal times, will decrease in the
same ratio.

PROPOSITION III. PROBLEM I.

To define the motion of a body which, in a similar medium, ascends or
descends in a right line, and is resisted in the ratio of its velocity, and
acted upon by an uniform force of gravity.

The body ascending, let the gravity be expound
ed by any given rectangle BACH ; and the resist
ance of the medium, at the beginning of the ascent,
by the rectangle BADE, taken on the contrary side
Jfl e B^l | L- of the right line AB. Through the point B, with
the rectangular asymptotes AC, CH, describe an
hyperbola, cutting the perpendiculars DE, de, ID

SEC. I.j OF NATURAL PHILOSOPHY. 253

G, g ; and the body ascending will in the time DGgd describe the space
EG«-e; in the time DGBA, the space of the whole ascent EGB ; in the
time ABK1, the space of descent BFK ; and in the time IKki the space of
descent KFfk; and the velocities of the bodies (proportional to the re
sistance of the medium) in these periods of time will be ABED, ABed, O,
ABFI, AB/z respectively ; and the greatest velocity which the body can
acquire by descending will be BACH.

For let the rectangle BACH be resolved into in
numerable rectangles AA', K/, Lm, M//, *fea, which
shall be as the increments of the velocities produced
in so many equal times; then will 0, AAr, AL Am, An,
ifec., be as the whole velocities, and therefore (by suppo
sition) as the resistances of the medium in the be-

ginning of each of the equal times. Make AC to AJLLB

AK, or ABHC to AB/vK, as the force of gravity to the resistance in the
beginning of the second time ; then from the force of gravity subduct the
resistances, and ABHC, K/vHC, L/HC, MwHC, (fee., will be as the abso
lute forces with which the body is acted upon in the beginning of each of
the times, and therefore (by Law I) as the increments of the velocities, that
is, as the rectangles AA-, K/, Lm, M//, (fee., and therefore (by Lem. 1, Book
II) in a geometrical progression. Therefore, if the right lines K£, LI
M/TO, N//, &c., are produced so as to meet the hyperbola in q, r, s, t, (fee..
the areas AB^K, K</rL, LrsM, MsJN, (fee., will be equal, and there
fore analogous to the equal times and equal gravitating forces. But the
area AB^K (by Corol. 3, Lem. VII and VIII, Book I) is to the area Bkq
as K^ to \kq, or AC to |AK, that is, as the force of gravity to the resist
ance in the middle of the first time. And by the like reasoning, the areas
<?KLr, rLMs, sMN/, (fee., are to the areas qklr, rims, smnt, (fee., as the
gravitating forces to the resistances in the middle of the second, third, fourth
time, and so on. Therefore since the equal areas BAKq, </KLr, rLMs,
sMN/, (fee., are analogous to the gravitating forces, the areas Bkq, qklr,
rims, smut, (fee., will be analogous to the resistances in the middle of
each of the times, that is (by supposition), to the velocities, and so to the
spaces described. Take the sums of the analogous quantities, and the areas
Bkq, Elr, Ems, But, (fee., will be analogous to the whole spaces described ;
and also the areas AB<?K, ABrL, ABsM, AB^N, (fee., to the times. There
fore the body, in descending, will in any time ABrL describe the space Blr,
and in the time Lr^N the space rlnt. Q,.E.D. And the like demonstra
tion holds in ascending motion.

COROL. 1. Therefore the greatest velocity that the body can acquire by
falling is to the velocity acquired in any given time as the £iven force ol
gravity which perpetually acts upon it to the resisting force which opposes
it at the end of that time.

854

THE MATHEMATICAL PRINCIPLES

[BOOK IL

COROL. 2. But the time being augmented in an arithmetical progression,
the sum of that greatest velocity and the velocity in the ascent, and also
their difference in the descent, decreases in a geometrical progression.

COROL. 3. Also the differences of the spaces, which are described in equal
differences of the times, decrease in the same geometrical progression.

COROL. 4. The space described by the body is the difference of two
spaces, whereof one is as the time taken from the beginning of the descent,
and the other as the velocity; which [spaces] also at the beginning of the
descent are equal among themselves.

PROPOSITION IV. PROBLEM II.

Supposing the force of gravity in any similar medium to be uniform,
and to tend perpendicularly to the plane of the horizon ; to define the
motion of a projectile therein, which suffers resistance proportional to
its velocity.

Let the projectile go from any place D in
the direction of any right line DP, and let
its velocity at the beginning of the motion
be expounded by the length DP. From the
point P let fall the perpendicular PC on the
horizontal line DC, and cut DC in A, so
that DA may be to AC as the resistance
of the medium arising from the motion up
wards at the beginning to the force of grav
ity; or (which comes to the same) so that
t ie rectangle under DA and DP may be to
that under AC and CP as the whole resist
ance at the beginning of the motion to the
force of gravity. With the asymptotes
DC, CP describe any hyperbola GTBS cut
ting the perpendiculars DG, AB in G and
B ; complete the parallelogram DGKC, and
let its side GK cut AB in Q,. Take

N in the same ratio to QB as DC is in to CP ; and from any point R of the
right line DC erect RT perpendicular to it, meeting the hy] erbola'in T,
and the right lines EH, GK, DP in I, t, and V ; in that perpendicular

take Vr equal to — ~- , or which is the same thing, take Rr equal to

(""""PIT?

— ^T— ; and the projectile in the time DRTG will arrive at the point r

describing the curve line DraF, the locus of the point r ; thence it will
come to its greatest height a in the perpendicular AB j and afterwards

SEC. 1.J » OF NATURAL PHILOSOPHY. 255

ever approach to the asymptote PC. And its velocity in any pjint r will
be as the tangent rL to the curve. Q.E.I.

For N is to Q,B as DC to CP or DR to RV, and therefore RV is equal to

PR X QB ,' -•..". "v v DRXQB-/GT

— ^r - , and R/' (that is, RV — Vr, or - -- ^ --- ) is equal to

DR X Ap — RDGT

-- ~ --- . JNow let the time be expounded by the area

RDGT and (by Laws, Cor. 2), distinguish the motion of the body into
two others, one of ascent, the other lateral. And since the resistance is as
the motion, let that also be distinguished into two parts proportional and
contrary to the parts of the motion : and therefore the length described by
the lateral motion will be (by Prop. II, Book II) as the line DR, and the
height (by Prop. Ill, Book II) as the area DR X AB — RDGT, that is,
as the line Rr. But in the very beginning of the motion the area RDGT
is equal to the rectangle DR X AQ, and therefore that line Rr (or
DRx AB —

that is, as CP to DC ; and therefore as the motion upwards to the motion
lengthwise at the beginning. Since, therefore, Rr is always as the height,
and DR always as the length, and Rr is to DR at the beginning as the
height to the length, it follows, that Rr is always to DR as the height to
the length ; and therefore that the body will move in the line DraF. which
is the locus of the point r. QJE.D.

DR X AB RDGT

COR. 1. Therefore Rr is equal to -- ^ ------ ^ - . and therefore

if RT be produced to X so that RX may be equal to -- ^ -- ; that is,

if the parallelogram ACPY be completed, and DY cutting CP in Z be
drawn, and RT be produced till it meets DY in X ; Xr will be equal to

RDGT

— ^ — , and therefore proportional to the time.

COR. 2. Whence if innumerable lines CR, or, which is the same, innu
merable lines ZX, be taken in a geometrical progression, there will be as
many lines Xr in an arithmetical progression. And hence the curve DraF
is easily delineated by the table of logarithms.

COR. 3. If a parabola be constructed to the vertex D, and the diameter
DG produced downwards, and its latus rectum is to 2 DP as the whole
resistance at the beginning of the notion to the gravitating force, the ve
locity with which the body ought *o go from the place D, in the direction
of the right line DP, so as in an uniform resisting medium to describe the
curve DraF, will be the same as that with which it ought to go from the
same place D in the direction of the same right line DP, so as to describe

256

THE MATHEMATICAL PRINCIPLES ~

[BOOK II

I a parabola in a non-resisting medium. For
the latus rectum of this parabola, at the very

DV2

beginning of the motion, is -y-', andVris

tGT DR x T*

-~JT- or £^T — . But a right line, which,

if drawn, would touch the hyperbola GTS in
K G, is parallel to DK, and therefore T* is

CKX DR

c

QBx DC

^ , and N is — ~pp • Ahd there-

DC

DR2 X CK x CP
fore Vr is equal to 2DC2 X QlT~; *^at *S (Because D^ an<* ^)C, DV

DV2 x CK ~x OP

and DP are proportionals), to — ^T5 Fcrr — J an<* tne ^atus reeturn

DV2

- comes out -

2DP2 X QB

are proportional),

CK X CP
2DP2 X DA

AC X CP

CP X AC ; that is, as the resistance to the gravity.

(because

and therefore is to 2DP as DP X DA to

Q.E.D.

COR. 4. Hence if a body be projected from
any place D with a given velocity, in the
direction of a right line DP given by posi
tion, and the resistance of the medium, at
the beginning of the motion, be given, the
curve DraF, which that body will describe,
may be found. For the velocity being
given, the latus rectum of the parabola is
given, as is well known. And taking 2DP
to that latus rectum, as the force of gravity
to the resisting force, DP is also given.
Then cutting DC in A, so that GP X AC
may be to DP X DA in the same ratio of
the gravity to the resistance, the point A
will be given. And hence the curve DraF
is also given.

COR. 5. And, on the contrary, if the
curve DraF be given, there will be given
x>th the velocity of the body and the resistance of the medium in each of
the places r. For the ratio of CP X AC to DP X DA being given, there
is given both the resistance of the medium at the beginning of the motion
and the latus rectum of the parabola ; and thence the velocity at the be
ginning of the motion is given also. Then from the length of the tangent

SEC. I.] OF NATURAL PHILOSOPHY. 257

L there is given both the velocity proportional to it, and the resistance
proportional to the velocity in any place r.

COR. 6. But since the length 2DP is to the latus rectum of the para
bola as the gravity to the resistance in D ; and, from the velocity aug
mented, the resistance is 'ti gmented in the same ratio, but the latus rectum
of the parabola is augmented in the duplicate of that ratio, it is plain thot
the length 2DP is augmented in that simple ratio only ; and is therefore
always proportional to the velocity ; nor will it be augmented or dimin
ished by the change of the angle CDP, unless the velocity be also changed.

COR. 7. Hence appears the method of deter
mining the curve DraF nearly from the phe-
nomena, and thence collecting the resistance and
velocity with which the body is projected. Let
two similar and equal bodies be projected with
the same velocity, from the place D, in differ
ent angles CDP, CDp ; and let the places F,
f. where they fall upon the horizontal plane
DC, be known. Then taking any length for D */ F

DP or Dp suppose the resistance in D to be to
the suavity in any ratio whatsoever, and let that

ratio be expounded by any length SM. Then, , _

by computation, from that assumed length DP, ^x

find the lengths DF, D/; and from the ratio

F/

-p^, found by calculation, subduct the same ratio as found by experiment ;

and let the cKfference be expounded by the perpendicular MN. Repeat the
same a second and a third time, by assuming always a new ratio SM of the
resistance to the gravity, and collecting a new difference MN. Draw the
affirmative differences on one side of the right line SM, and the negative
on the other side ; and through the points N, N, N, draw a regular curve
NNN. cutting the right line SMMM in X, and SX will be the true ratio
of the resistance to the gravity, which was to be found. From this ratio
the length DF is to be collected by calculation ; and a length, which is to
the assumed length DP as the length DF known by experiment to the
length DF just now found, will be the true length DP. This being known,
you will have both the curve line DraF which the body describes, and also
the velocity and resistance of the body in each place.

SCHOLIUM.

But, yet, that the resistance of bodies is in the ratio of the velocity, is more
a mathematical hypothesis than a physical one. In mediums void of all te
nacity, the resistances made to bodies are in the duplicate ratio of the ve
locities. For by the action of a swifter body, a greater motion in propor-

17

THE MATHEMATICAL PRINCIPLES [BoOK IL

tion to a greater velocity is communicated to the same quantity of the
medium in a less time ; and in an equal time, by reason of a greater quan
tity of the disturbed medium, a motion is communicated in the duplicate
ratio greater ; and the resistance (by Law II and III) is as the motion
communicated. Let us, therefore, see what motions arise from this law of
resistance.

SECTION II.

'If the motion of bodies that are resisted in tfie duplicate ratio of their

velocities.

PROPOSITION V. THEOREM III.

Ff a body is resisted in the duplicate ratio of its velocity, and moves by
its innate force only through a similar medium; and the times be
taken in a geometrical progression, proceeding from less to greater
terms : I say, that the velocities at the beginning of each of the times
are in the same geometrical progression inversely ; and that the spaces
are equal, which are described in each of the times.
For since the resistance of the medium is proportional to the square of
the velocity, and the decrement of the velocity is proportional to the resist
ance : if the time be divided into innumerable equal particles, the squares of
the velocities at the beginning of each of the times will be proportional to
the differences of the same velocities. Let those particles of time be AK,
KL, LM, &c., taken in the right line CD; and
erect the perpendiculars AB, Kk, L/, Mm, &c.,
meeting the hyperbola BklmG, described with the
centre C, and the rectangular asymptotes CD, CH.
in B, kj I, m, (fee. ; then AB will be to Kk as CK
to CA, and, by division, AB — Kk to Kk as AK

C ARIMT to ^A> an(1 alternate^ AB — ^C to AK as Kk

to CA ; and therefore as AB X Kk to AB X CA.

Therefore since AK and AB X CA are given,* AB — Kk will be as AB
X Kk ; and, lastly, when AB and KA* coincide, as AB2. And, by the like
reasoning, KAr-U, J J-M/??, (fee., will be as Kk2. LI2, (fee. Therefore the
squares of the lines AB, KA", L/, Mm, (fee., are as their differences ; and,
therefore, since the squares of the velocities were shewn above to be as their
differences, the progression of both will be alike. This being demonstrated
it follows also that the areas described by these lines are in a like progres
sion with the spaces described by these velocities. Therefore if the velo
city at the beginning of the first time AK be expounded by the line AB,

SEC. II.] OF NATURAL PHILOSOPHY. 25CJ

and the velocity at the beginning of the second time KL by the line K&
and the length described in the hrst time by the area AKA*B, all the fol
lowing velocities will be expounded by the following lines \J. Mm, .fee.
and the lengths described, by the areas K/, I mi. &c. And, by compo
sition, if the whole time be expounded by AM, the sum of its parts, the
whole length described will be expounded by AM/ftB the sum of its parts.
Now conceive the time AM to be divided into the parts AK, KL, LM, (fee
so that CA, CK, CL, CM, (fee. may be in a geometrical progression ; and
those parts will be in the same progression, and the velocities AB, K/r,
L/, Mm, (fee., will be in the same progression inversely, and the spaces de
scribed Ak, K/, Lw, (fee., will be equal. Q,.E.D.

COR. 1. Hence it appears, that if the time be expounded by any part
AD of the asymptote, and the velocity in the beginning of the time by the
ordinate AB, the velocity at the end of the time will be expounded by the
ordinate DG ; and the whole space described by the adjacent hyperbolic
area ABGD ; and the space which any body can describe in the same time
AD, with the first velocity AB, in a non-resisting medium, by the rectan
gle AB X AD.

COR 2. Hence the space described in a resisting medium is given, by
taking it to the space described with the uniform velocity AB in a non-
resisting medium, as the hyperbolic area ABGD to the rectangle AB X AD.

COR. 3. The resistance of the medium is also given, by making it equal,
in the very beginning of the motion, to an uniform centripetal force, which
could generate, in a body falling through a non-resisting medium, the ve
locity AB in the time AC. For if BT be drawn touching the hyperbola
in B. and meeting the asymptote in T, the right line AT will be equal to
AC, and will express the time in which the first resistance, uniformly con
tinned, may take away the whole velocity AB

COR. 4. And thence is also given the proportion of this resistance to the
force of gravity, or ay other given centripetal force.

COR. 5. And, vice versa, if there is given the proportion of the resist-
; nee to any given centripetal force, the time AC is also given, in which c
centripetal force equal to the resistance may generate any velocity as AB ;
and thence is given the point B. through which the hyperbola, having CH
CD for its asymptotes, is to be described : as also the space ABGD, which a
body, by beginning its motion with that velocity AB, can describe in any
time AD. in a similar resisting medium.

PROPOSITION VI. THEOREM lVrc

Homogeneous and equal spherical bodies, opposed hy resistances that are
in the duplicate ratio of the velocities, and moving on by their innate
force only, will, in times which are reciprocally as the velocities at thr.

260 THE MATHEMATICAL PRINCIPLES |BOOK II,

v

beg-in fiing, describe equal spaces, and lose parts of their velocities pro
portional to the wholes.

To the rectangular asymptotes CD, CH de
scribe any hyperbola B6Ee, cutting the perpen
diculars AB, rib, DE, de in B, b, E, e; let the
initial velocities be expounded by the perpendicu
lars AB, DE, and the times by the lines Aa, Drf.
Therefore as Aa is to l)d, so (by the hypothesis)
. is DE to AB, and so (from the nature of the hy-
C "^ perbola) is CA to CD ; and, by composition, so is

Crt to Cd. Therefore the areas ABba, DEerf, that is, the spaces described,
are equal among themselves, and the first velocities AB, DE are propor
tional to the last ab, de ; and therefore, by division, proportional to the
parts of the velocities lost, AB — ab, DE — de. Q.E.D.

PROPOSITION VII. THEOREM V.

If spherical bodies are resisted in the duplicate ratio of their velocities,
in times which are as the first motions directly, and the first resist
ances inversely, they will lose parts of their motions proportional to the
wholes, and will describe spaces proportional to those times and the
first velocities conjunct It/.

For the parts of the motions lost are as the resistances and times con
junctly. Therefore, that those parts may be proportional to the wholes,
the resistance and time conjunctly ought to be as the motion. Therefore the
time will be as the motion directly and the resistance inversely. Where
fore the particles of the times being taken in that ratio, the bodies will
always loso parts of their motions proportional to the wholes, and there
fore will retain velocities always proportional to their first velocities.
And because of the given ratio of the velocities, they will always describe
spaces which are as the first velocities and the times conjunctly. Q.E.D.
COR. 1. Therefore if bodies equally swift are resisted in a duplicate ra
tio of their diameters, homogeneous globes moving with any velocities
whatsoever, by describing spaces proportional to their diameters, will lose
parts of their motions proportional to the wholes. For the motion of each
o-lobe will be as its velocity and mass conjunctly, that is, as the velocity
and the cube of its diameter ; the resistance (by supposition) will be as the
square of the diameter and the square of the velocity conjunctly ; and the
•time (by this proposition) is in the former ratio directly, and in the latter
inversely, that is, as the diameter directly and the velocity inversely ; and
therefore' the space, which is proportional to the time and velocity is as
the diameter.

COR. 2. If bodies equally swift are resisted in a sesquiplicate ratio of
their diameters, homogeneous globes, moving with any velocities whatso-

SEC. 1L] OF NATURAL PHILOSOPHY. 261

ever, by describing spaces that are in a sesquiplicate ratio of the diameters,
will lose parts of their motions proportional to the wholes.

COR. 3. And universally, if equally swift bodies are resisted in the ratio
of any power of the diameters, the spaces, in which homogeneous globes,
moving with any velocity whatsoever, will lose parts of their motions pro
portional to the wholes, will be as the cubes of the diameters applied to
that power. Let those diameters be D and E : and if the resistances, where
the velocities are supposed equal, are as T)n and E" ; the spaces in which
the globes, moving with any velocities whatsoever, will lose parts of their
motions proportional to the wholes, will be as D3 — n and E3 — n. And
therefore homogeneous globes, in describing spaces proportional to D 3 — n
and E3 — n, will retain their velocities in the same ratio to one another as
at the beginning.

COR. 4. Now if the globes are not homogeneous, the space described by
the denser globe must be augmented in the ratio of the density. For the
motion, with an equal velocity, is greater in the ratio of the density, and
the time (by this Prop.) is augmented in the ratio of motion directly, and
the space described in the ratio of the time.

COR. 5. And if the globes move in different mediums, the space, in a
medium which, cccteris paribus, resists the most, must be diminished in the
ratio of the greater resistance. For the time (by this Prop.) will be di
minished in the ratio of the augmented resistance, and the space in the ra
tio of the time.

LEMMA II.

The moment of any genitum is equal to the moments of each of the gen-
eratinrr sides drawn into the indices of the powers of those sides, and
into their co-efficients continually.

I call any quantity a genitum which is not made by addition or sub-
duction of divers parts, but is generated or produced in arithmetic by the
multiplication, division, or extraction of the root of any terms whatsoever :
in geometry by the invention of contents and sides, or of the extremes and
means of proportionals. Quantities of this kind are products, quotients,
roots, rectangles, squares, cubes, square and cubic sides, and the like.
These quantities I here consider as variable and indetermined, and increas
ing or decreasing, as it were, by a perpetual motion or flux ; and I under
stand their momentaneous increments or decrements by the name of mo
ments ; so that the increments may be esteemed as added or affirmative
moments ; and the decrements as subducted or negative ones. But take
care not to look upon finite particles as such. Finite particles are not
moments, but the very quantities generated by the moments. We are to
conceive them as the just nascent principles of finite magnitudes. Nor do
we in this Lemma regard the magnitude of the moments, but their firsf

262 THE MATHEMATICAL PRINCIPLES [BoOK 11

proportion, as nascent. It will be the same thing, if, instead of moments,
we use either the velocities of the increments and decrements (which may
also be called the motions, mutations, and fluxions of quantities), or any
finite quantities proportional to those velocities. The co-efficient of any
generating side is the quantity which arises by applying the genitum to
ihat side.

Wherefore the sense of the Lemma is, that if the moments of any quan
tities A, B, C, &c., increasing or decreasing by a perpetual flux, or the
velocities of the mutations which are proportional to them, be called a, 6,
r, (fee., the moment or mutation of the generated rectangle AB will be «B
-h b A ; the moment of the generated content ABC will be aBC -f b AC 4

-1 -2. .1

cAB; and the moments of the generated powers A2. A3, A4, A2, A2. A3,
A*, A — ', A — 2, A— * will be 2aA, 3a A2, 4aA3, |«A — *, f«A*

11 3

i« A — s, |/iA — 3, — a A — 2, — 2aA — 3, — £aA — 2 respectively; and
in general, that the moment of any power A^, will be ^ aAn-^. Also,
that the moment of the generated quantity A2B will be 2aAB 4- bA~ ; the
moment of the generated quantity A 3B4C2 will be 3«A2B4C2 + 4/>A3

A3
B3C2 4-2cA3B'C; and the moment of the generated quantity — or

A »B — 2 will be 3aA 2 B — 2 — 2bA 3B — 3 ; and so on. The Lemma is
thus demonstrated.

CASE 1. Any rectangle, as AB, augmented by a perpetual flux, when, as
yet, there wanted of the sides A and B half their moments \a and \b, was
A — \a into B — \b, or AB — ±a B — \b A + \ab ; but as soon as the
sides A and B are augmented by the other half moments, the rectangle be
comes A 4- 4-a into B 4- \b, or AB -f ^a B 4- \b A -f \ab. From this
rectangle subduct the former rectangle, and there will remain the exces.?
aE -f bA. Therefore with the whole increments a and b of the sides, tin
increment aB + f>A of the rectangle is generated. Q.K.D.

CASE 2. Suppose AB always equal to G, and then the moment of the
content ABC or GC (by Case 1) will be^C + cG, that is (putting AB and
aB + bA for G and «*), aBC -h bAC 4- cAB. And the reasoning is the
same for contents under ever so many sides. Q.E.D.

CASE 3. Suppose the sides A, B, and C, to be always equal among them
selves; and the moment «B + />A, of A2, that is, of the rectangle AB,
will be 2aA ; and the moment aBC + bAC + cAB of A3, that is, of the
content ABC, will be 3aA2. And by the same reasoning the moment of
any power An is naAn — '. Q.E.D

CASE 4. Therefore since -7 into A is 1, the moment of -r- drawn into
A A

SEC. 11.] OF NATURAL PHILOSOPHY. 263

A, together with — drawn into a. will be the moment of 1, that is, nothing.
A

Therefore the moment of -r, or of A — ', is -r— . And generally since

A .A

T- into An is I, the moment of —drawn into An together with — into
A n A. A n

naA" — ! will be nothing. And, therefore, the moment of -r- or A — n

A

will be— £T^~7- Q-E.D.

V . t . ' i

CASE 5. And since A2 into A2 is A, the moment of A1 drawn into 2 A2

will be a (by Case 3) ; and, therefore, the moment of A7 will be n~r~r or

^A-j

£#A — £. And, generally, putting A~^ equal to B, then Am will be equal
to Bn, and therefore maAm — ! equal to nbBn — ' , and ma A — ' equal to

?tbB — ', or tib A — ^ 5 an<i therefore ri a A —^~ is equal to &, that is, equal

to the moment of A^. Q.E.D.

CASE 6. Therefore the moment of any generated quantity AmBn is the
moment of Am drawn into Bn, together with the moment of Bn drawn into
A", that is, maAm — ' B" -f- nbBn — ! Am; and that whether the indices
in arid n of the powers be whole numbers or fractions, affirmative or neg
ative. And the reasoning is the same for contents under more powers.
Q.E.D.

COR. 1. Henoe in quantities continually proportional, if one term is
given, the moments of the rest of the terms will be as the same terms mul
tiplied by the number of intervals between them and the given term. Let
A, B, C, D; E, F, be continually proportional ; then if the term C is given,
the moments of the rest of the terms will be among themselves as — 2A,
— B? D, 2E, 3F.

COR. 2. And if in four proportionals the two means are given, the mo
ments of the extremes will be as those extremes. The same is to be un
derstood of the sides of any given rectangle.

COR. 3. And if the sum or difference of two squares is given, the mo
ments of the sides will be reciprocally as the sides.

SCHOLIUM.

In a letter of mine to Mr. /. Collins, dated December 10, 1672, having
described a method of tangents, which I suspected to be the same with
Slusius*s method, which at that time wag not made public, I subjoined these
words • This is one particular, or rather a Corollary, of a general nte

264 THE MATHEMATICAL PRINCIPLES [BjOK II.

thod, which extends itself, without any troublesome calculation, not ojdy
to the drawing of tangents to any curve lines, whether geometrical or
mechanical, or any how respecting right lines or other cnrves, but also
to the resolving other abstrnser kinds of problems about the crookedness,
areas, lengths, centres of gravity of curves, &c. ; nor is it (as Hudd^ri's
method de Maximis & Minimia) limited to equations which are free from
surd quantities. This method I have interwoven with that other oj
working in equations, by reducing them to infinite serie?. So far that
letter. And these last words relate to a treatise I composed on that sub
ject in the year 1671. The foundation of that general method is contain
ed in the preceding Lemma.

PROPOSITION VIII. THEOREM VI.

If a body in an uniform medium, being uniformly acted upon by the force
of gravity, ascends or descends in a right line ; and the whole space
described be distinguished into equal parts, and in the beginning of
each of the parts (by adding or subducting the resisting force of the
medium to or from the force of gravity, when the body ascends or de
scends] yon collect the absolute forces ; I say, that those absolute forces
ire in a geometrical progression.

For let the force of gravity be expounded by the
given line AC ; the force of resistance by the indefi
nite line AK ; the absolute force in the descent of the
body by the difference KC : the velocity of the I tody
<^LKJL&i>F/ by a line AP, which shall be a mean proportional be
tween AK and AC, and therefore in a subduplicate ratio of the resistance;
the increment of the resistance made in a given particle of time by the li-
neola KL, and the contemporaneous increment of the velocity by the li-
neola PQ ; and with the centre C, and rectangular asymptotes CA, CH,
describe any hyperbola BNS meeting the erected perpendiculars AB, KN,
LO in B, N and O. Because AK is as AP2 , the moment KL of the one will
be as the moment 2APQ of the other, that is, as AP X KC ; for the in
crement PQ of the velocity is (by Law II) proportional to the generating
force KC. Let the ratio of KL be compounded with the ratio KN, and
the rectangle KL X KN will become as AP X KC X KN ; that is (because
the rectangle KC X KN is given), as AP. But the ultimate ratio of the
hyperbolic area KNOL to the rectangle KL X KN becomes, when the
points K and L coincide, the ratio of equality. Therefore that hyperbolic
evanescent area is as AP. Therefore the whole hyperbolic area ABOL
is composed of particles KNOL which are always proportional to the
velocity AP; and therefore is itself proportional to the space described
with that velocity. Let ,that area be now divided into equal parts

SEC. IJ.J

OF NATURAL PHILOSOPHY.

265

as ABMI, IMNK, KNOL, (fee., and the absolute forces AC, 1C, KC, LC,

(fee., will be in a geometrical progression. Q,.E.D. And by a like rea
soning, in the ascent of the body, taking, on the contrary side of the point
A, the equal areas AB?m, i/nnk, knol, (fee., it will appear that the absolute
forces AC. iG, kC, 1C, (fee., are continually proportional. Therefore if all
the spaces in the ascent and descent are taken equal, all the absolute forces
1C, kC, iC, AC, 1C, KC, LC, (fee., will be continually proportional. Q,.E.D.

COR. 1. Hence if the space described be expounded by the hyperbolic
area ABNK, the force of gravity, the velocity of the body, and the resist
ance of the medium, may be expounded by the lines AC, AP, and AK re
spectively • and vice versa.

COR. 2. And the greatest velocity which the body can ever acquire in
an infinite descent will be expounded by the line AC.

COR. 3. Therefore if the resistance of the medium answering to any
given velocity be known, the greatest velocity will be found, by taking it
to that given velocity in a ratio subduplicate of the ratio which the force
of gravity bears to that known resistance of the medium.

PROPOSITION IX. THEOREM VII.

Supposing ivhat is above demonstrated, I say, that if the tangents of t-he
angles of the sector of a circle, and of an hyperbola, be taken propor
tional to the velocities, the radius being of a fit magnitude, all the time
of the ascent to the highest place icill be as the sector of the circle, and
all the time of descending from the highest place as the sector of t/ie
hyperbola.

To the right line AC, which ex
presses the force of gravity, let AD
drawn perpendicular and equal. From
the centre D with the semi-diameter
AD describe as well the cmadrant A^E

-t

of a circle, as the rectangular hyper
bola AVZ, whose axis is AK, principal
vertex A, and asymptote DC. Let Dp,
DP be drawn ; and the circular sector
AtD will be as all the time of the as
cent to the highest place ; and the hy
perbolic sector ATD as all the time of descent from the highest place; ii
BO be that the tangents Ap, AP of those sectors be as the velocities.

CASE 1. Draw Dvq cutting off the moments or least particles tDv and
^ ?, described in the same time, of the sector ADt and of the triangle
AD/?. Since those particles (because of the common angle D) are in a du
plicate ratio of the sides, the particle tDv will be as — -^-^-— , that is

266

THE MATHEMATICAL PRINCIPLES

[BOOK li.

(because tD is given), as ^f. But joD8 is AD3 + Ap2, that is, AD2 -h
AD X AA-, or AD X Gk ; and (/Dp is 1 AD X pq. Therefore tDv, the

BO

particle of the sector, is as ^ , ; that is, as the least decrement pq of the

velocity directly, and the force Gk which diminishes the velocity, inversely ;
and therefore as the particle of time answering to the decrement of the ve
locity. And, by composition, the sum of all the particles tDv in the sector
AD/ will be as the sum of the particles of time answering to each of the
lost particles pq of the decreasing velocity Ap, till that velocity, being di
minished into nothing, vanishes; that is, the whole sector AD/ is as the
whole time of ascent to the highest place. Q.E.D.

CASE 2. Draw DQV cutting off the least particles TDV and PDQ of
the sector DAV, and of the triangle DAQ ; and these particles will be to
each other as DT2 to DP2, that is (if TX and AP are parallel), as DX2
to DA2 or TX2 to AP2 ; and, by division, as DX2 — TX2 to DA2 -
AP2. But. from the nature of the hyperbola, DX2 — TX2 is AD2 ; and, by
the supposition, AP2 is AD X AK. Therefore the particles are to each
other as AD2 to AD2 — AD X AK ; that is, as AD to AD — AK or AC

; and

to CK : and therefore the particle TDV of the sector is -

PQ

therefore (because AC and AD are given) as

CK

that is, as the increment

of the velocity directly, and as the force generating the increment inverse
ly ; and therefore as the particle of the time answering to the increment.
And, by composition, the sum of the particles of time, in which all the par
ticles PQ of the velocity A I3 are generated, will be as the sum of the par
ticles of the sector ATI) ; that is, the whole time will be as the whole
sector. Q.E.D.

COR. 1. Hence if AB be equal to a
fourth part of AC, the space which a body
will describe by falling in any time will
be to the space which the body could de
scribe, by moving uniform]}' on in the
same time with its greatest velocity
AC, as the area ABNK, which ex
presses the space described in falling to
the area ATD, which expresses the
time. For since AC is to AP as AP
_ to AK, then (by Cor. 1, Lem. II, of this

Book) LK is to PQ as 2AK to AP, that is, as 2AP to AC, and thence
LK is to ^PQ as AP to JAC or AB ; and KN is to AC or AD as AB tc

. II.] OF NATURAL PHILOSOPHY. 267

UK ; and therefore, ex cequo, LKNO to DPQ, as AP to CK. But DPQ
was to DTV as CK to AC. Therefore, ex aquo, LKNO is to DTV r,?
AP to AC ; that is, as the velocity of the falling body to the greatest
velocity which the body by falling can acquire. Since, therefore, the
moments LKNO and DTV of the areas ABNK and ATD are as the ve
locities, all the parts of those areas generated in the same time will be as
the spaces described in the same time ; and therefore the whole areas ABNK
and ADT, generated from the beginning, will be as the whole spaces de
scribed from the beginning of the descent. Q.E.D.

COR. 2. The same is true also of the space described in the ascent.
That is to say, that all that space is to the space described in the same
time, with the uniform velocity AC, as the area ABttk is to the sector ADt.

COR. 3. The velocity of the body, falling in the time ATD, is to the
velocity which it would acquire in the same time in a non-resisting space,
as the triangle APD to the hyperbolic sector ATD. For the velocity in
a non-resisting medium Avould be as the time ATD, and in a resisting me
dium is as AP, that is, as the triangle APD. And those velocities, at the
beginning of the descent, are equal among themselves, as well as those
areas ATD, APD.

COR. 4. By the same argument, the velocity in the ascent is to the ve
locity with which the body in the same time, in a non-resisting space, would
lose all its motion of ascent, as the triangle ApD to the circular sector
AtD ; or as the right line Ap to the arc At.

COR. 5. Therefore the time in which a body, by falling in a resisting
medium, would acquire the velocity AP, is to the time in which it would
acquire its greatest velocity AC, by falling in a non-resisting space, as the
sector ADT to the triangle ADC : and the time in which it would lose its
velocity Ap, by ascending in a resisting medium, is to the time in which
it would lose the same velocity by ascending in a non-resisting space, as
the arc At to its tangent Ap.

COR. 6. Hence from the given time there is given the space described in
the ascent or descent. For the greatest velocity of a body descending in
wfinitum is given (by Corol. 2 and 3, Theor. VI, of this Book) ; and thence
the time is given in which a body would acquire that velocity by falling
in a non-resisting space. And taking the sector ADT or ADt to the tri
angle ADC in the ratio of the given time to the time just now found,
there will be given both the velocity AP or Ap, and the area ABNK or
AB//&, which is to the sector ADT, or AD/, as the space sought to the
space which would, in the given time, be uniformly described with that
greatest velocity found just before.

COR. 7. And by going backward, from the given space of ascent or de
scent AB»?A: or ABNK, there will be given the time AD* or ADT.

268

THE MATHEMATICAL PRINCIPLES

[BOOK ii

PROPOSITION X. PROBLEM III.

Suppose the uniform force of gravity to tend directly to the plane of the
horizon, and the resistance to be as the density of the medium and the
square of the velocity coiijunctly : it is proposed to find the density of
the medium in each place, which shall make the body move in any
given carve line ; the velocity of the body and the resistance of the
medium in each place.

Let PQ be a plane perpendicular to
the plane of the scheme itself; PFHQ
a curve line meeting that plane in the
points P and Q ; G, H, I, K four
places of the body going on in this
\ curve from F to Q ; and GB, HC, ID,
KE four parallel ordinates let fall

p A. 33 c^D E Q from these points to the horizon, and

standing on the horizontal line PQ at the points B, C, D, E ; and let the
distances BC, CD, DE, of the ordinates be equal among themselves. From
the points G and H let the right lines GL, HN, be drawn touching the
curve in G and H, and meeting the ordinates CH, DI, produced upwards,
in L and N : and complete the parallelogram HCDM. And the times in
which the body describes the- arcs GH, HI, will be in a subduplicate ratio
of the altitudes LH, NI; which the bodies would describe in those times,
by falling from the tangents; and the velocities will be as the lengths de
scribed GH, HI directly, and the times inversely. Let the times be ex-

C*TT TTT
pounded by T and t, and the velocities by —•=- and ---; and the decrement

J_ L

/-^TT TTT

of the velocity produced in the time t will be expounded by -7^ — .

This decrement arises from the resistance which retards the body, and from
the gravity which accelerates it. Gravity, in a falling body, which in its
fall describes the space NI, produces a velocity with which it would be able
to describe twice that space in the same time, as Galileo has demonstrated ;

2NI

that is, the velocity

: but if the body describes the arc HI, it augments
MIxNl

HI

— ; and therefore generates

that arc only by the length HI — HN or

only the velocity — iff—- Let this velocity be added to the before-

t X H.JL

mentioned decrement, and we shall have the decrement of the velocity

GH HI SMI X Nl

arising from the resistance alone, that is, -^ : — h

T

SEC. II.J

OF NATURAL PHILOSOPHY.

269

2NI.

Therefore since, in the same time, the action of gravity generates, in a fall
ing body, the velocity , the resistance will be to the gravity as 7^

HI 2MI X NI 2NI t X GH 2MI X NI

— + — TTT- to — or as — ^ — — HI -f

Now for the abscissas CB, CD,
CE, put — o, o, 2o. For the ordinate
CH put P ; and for MI put any series
Qo + Ro2 + So3 +, &c. And all
the terms of the series after the first,
that is, Ro2 + So3 +, (fee., will be
NI ; and the ordinates DI, EK, and

BGwill be P — Qo—Ro2— So3— p A B c T> E

(fee., P — 2Qo— 4Ro2— SSo3— , (fee., and P -\- Qo — Ro2 + So3—,
(fee., respectively. And by squaring the differences of the ordinates BG —
CH and CH — DI, and to the squares thence produced adding the squares
of BC and CD themselves, you will have oo -f- QQoo — 2QRo3 +, (fee.,
and oo -f QQoo -f 2QRo3 +, (fee., the squares of the arcs GH, HI ; whose

QRoo QRoo

roots o y - • , and o </! 4- QQ 4- — are the

1 + QQ v/1 + QQ s/1 -f QQ

arcs GH and HI. Moreover, if from the ordinate CH there be subducted
half the sum of the ordinates BG and DI, and from the ordinate DI there
be subducted half the sum of the ordinates CH and EK, there will remain
Roo and Roo + 3So3, the versed sines of the arcs GI and HK. And these
are proportional to the lineolae LH and NI, and therefore in the duplicate

ratio of the infinitely small times T and t : and thence the ratio ~, is ^

R + SSo R -f

^ or

So , t X GH TTT 2MI X NI ,

— : and — ^ HI H TTT , by substituting

R T HI

the values of ™, GH, HI, MI and NI just found, becomes -^-

J- /w-Lt/

I + QQ. Arid since 2NI is 2Roo, the resistance will be now to the

OO

gravity as -- TT'Q to 2Roo> that is> as 3S r to 4RR.

And the velocity will be such, that a body going off therewith from any
place H, in the direction of the tangent HN, would describe, in vacuo, a

parabola, whose diameter is HC, and its latus rectum NT or -- ^ ---- .

And the resistance is as the density of the medium and the square of
the velocity conjunctly ; and therefore the density of the medium is as the
resistance directly, and the square of the velocity inversely ; that is, as

270 THE MATHEMATICAL PRINCIPLES [BOOK II.

QQ __

4RR

Q.E.I.

COR. 1. If the tangent HN be produced both ways, so as to meet any

HT
ordinatc AF in T - will be equal to V/T+ QQ; and therefore in what

has gone before may be put for ^ \ -\- QQ. By this means the resistance
will be to the gravity as 3S X HT to 4RR X AC ; the velocity will be a*

•r-pj — --^, and the density of the medium will be as „— -TT-n.
AO -v/ ±i Jti X H 1

COR. 2. And hence, if the curve line PFHQ be denned by the relation
between the base or abscissa AC and the ordinate CH, as is usual, and the
value of the ordinate be resolved into a converging series, the Problem
will be expeditiously solved by the first terms of the series ; as in the fol
lowing examples.

EXAMPLE 1. Let the line PFHQ, be a semi-circle described upon the
diameter PQ, to find the density of the medium that shall make a projec
tile move in that line.

Bisect the diameter PQ in A ; and call AQ, n ; AC, a ; CH, e ; and
CD, o ; then DI2 or AQ,2 — AD 2 = nn — aa — 2ao — oo, or ec. — 2ao
— oo ; and the root being extracted by our method, will give DI = e —
ao oo aaoo ao3 a3 o3
~e~~~2e 2e? ~~~ W ~2? ' &C* Here put nn f°r ee + aa> and

ao nnoo anno3

DI will become = e — , &c.

e 2e3 2e5

Such series I distinguish into successive terms after this manner : I call
that the first term in which the infinitely small quantity o is not found ;
the second, in which that quantity is of one dimension only ; the third, in
which it arises to two dimensions ; the fourth, in which it is of three ; and
so ad infinitum. And the first term, which here is e, will always denote
the length of the ordinate CH, standing at the beginning of the indefinite

quantity o. The second term, which here is — , will denote the difference

between CH and DN ; that is, the lineola MN which is cut off by com
pleting the parallelogram HC DM; and therefore always determines the

CM?

position of the tangent HN ; as, in this case, by taking MN to HM as —

G

to o, or a to e. The third term, which here is -£—, will represent the li
neola IN, which lies between the tangent and the curve ; and therefore
determines the angle of contact IHN, or the curvature which the curve line

SEC. II.] OF NATURAL PHILOSOPHY. 271

has in H. If that lineola IN is of a finite magnitude, it will be expressed
by the third term, together with those that follow in wfinitu:.:i. But if
that lineola be diminished in. infini-
tnm, the terms following become in
finitely less than the third term, and
therefore may be neglected. The
fourth term determines the variation
of the curvature ; the fifth, the varia
tion of the variation ; and so on.

Whence, by the way, appears no con-p~" ~K B~C~D~E~

temptible use of these series in the solution of problems that depend upon
tangents, and the curvature of curves.

ao 77/700 anno 3

Now compare the series e — ^— ^~ &c., with the

e Ze3 Ze*

series P — Qo — - Roo — So3 — &c., and for P, Q, II and S? put e, -, ^-^

and ~ — , and for ^ 1 + QQ put 1 H or - ; and the density oi

2e 5 ' ee e '

the medium will come out as — ; that is (because n is given), as - or

lie ' e

~Yj, that is, as that length of the tangent HT, which is terminated at the
OH.

semi-diameter AF standing perpendicularly on PQ : and the resistance
will be to the gravity as 3a to 2>/, that is, as SAC to the diameter PQ of
the circle; and the velocity will be as i/ CH. Therefore if the body goes
from the place F, with a due velocity, in the direction of a line parallel to
PQ, and the density of the medium in each of the places H is as the length
of the tangent HT, and the resistance also in any place H is to the force
of gravity as SAC to PQ, that body will describe the quadrant FHQ of a
circle. Q.E.I.

But if the same body should go-frorn the place P, in the direction of a
line perpendicular to PQ, and should begin to move in an arc of the semi
circle PFQ, we must take AC or a on the contrary side of the centre A ;
and therefore its sign must be changed, and we must put — a for + a.

Then the density of the medium would come out as . But nature

does not admit of a negative density, that is, a density which accelerates
the motion of bodies; and therefore it cannot naturally come to pass that
a body by ascending from P should describe the quadrant PF of a circle.
To produce such an effect, a body ought to be accelerated by an impelling
medium, and not impeded by a resisting one.

EXAMPLE 2. Let the line PFQ be a parabola, having its axis AF per-

272

THE MATHEMATICAL PRINCIPLES

[BOOK IL

pendicular to the horizon PQ, to find the density of the medium, which
will make a projectile move in that line.

From the nature of the parabola, the rectangle PDQ,
is equal to the rectangle under the ordinate DI and some
given right line ; that is, if that right line be called b ;
PC, a; PQ, c; CH, e; and CD, o; the rectangle a
A. CD ~Q + o into c — a — o or ac — aa — 2ao -{-co — oo, ia

ac — aa

equal to the rectangle b into DI, and therefore DI is equal to -- 7 -- h

c — 2a oo c — 2a

— -. — o — —r. Now the second term — -, — o of this series is to be put
b b b

oo

for Q,o, and the third term -r for Roo. But since there are no more

terms, the co-efficient S of the fourth term will vanish ; and therefore the

S
ouantitv - — , to which the density of the medium is proper-

'

R v i

tional, will be nothing. Therefore, where the medium is of no density,

the projectile will move in a parabola ; as Galileo hath heretofore demon

strated. Q.E.I.

EXAMPLE 3. Let the line AGK be an hyperbola, having its asymptote

NX perpendicular to the horizontal plane AK, to find the density of the

medium that will make a projectile move in that line.

Let MX be the other asymptote, meeting
the ordinate DG produced in V ; and from
the nature of the hyperbola, the rectangle of
XV into VG will be given. There is also
given the ratio of DN to VX, and therefore
the rectangle of DN into VG is given. Let
that be bb : and, completing the parallelo
gram DNXZ, let BN be called a; BD, o;
NX, c; and let the given ratio of VZ to

ZX or DN be -. Then DN will be equal

MA. BD K N

bb
to a — o} VG equal to

, VZ equal to — X a — o. and GD or NX
a — o' n

m

m

-VZ— VG equal to c— — a + — o — . Let the term - — be

n n a — o a — o

bb bb bb bb ,

resolved into the converging series ~^" + ^° + ^l00 + ^4 °> &c'» and

m bb m bb bb bb

GD will become equal to c — - a— - + -o — ~ o — ^ o2 — 51 °

SEC. II.]

OF NATURAL PHILOSOPHY.

273

&c. The second term — o o of this series is to be u?ed for Qo; the

n aa

third — o2, with its sign changed for Ro2 ; and the fourth — o3, with its

m bb bb bb

sign changed also for So3, and their coefficients , — and — are to

n aa a a

be pat for Q, R, and S in the former rule. Which being done, the den-

bb
a*

sity of the medium will come out as , ,

bb

a

mm
nn

2mbb
naa

I

mm

-— , that is, if in VZ you take VY equal to

aa

aa

1 m2

VG, as YT7- For aa and —^ a 2

2mbb b'
nn n aa

2mbb b 4

— • H are the squares of XZ

n aa

and ZY. But the ratio of the resistance to gravity is found to be that of
3XY to 2YG ; and the velocity is that with which the body would de-

XY2

scribe a parabola, whose vertex is G, diameter DG, latus rectum ^v . Sup
pose, therefore, that the densities of the medium in each of the places G
are reciprocally as the distances XY, and that the resistance in any place
G is to the gravity as 3XY to 2YG ; and a body let go from the place A,
with a due velocity, will describe that hyperbola AGK. Q.E.I.

EXAMPLE 4. Suppose, mdeMtely, the line AGK to be an hyperbola
described with the centre X, and the asymptotes MX, NX, so that, having
constructed the rectangle XZDN, whose side ZD cuts the hyperbola in G
and its asymptote in V, VG may be reciprocally as any power DNn of the
line ZX or DN, whose index is the number n : to find the density of the
medium in which a projected body will describe this curve.

For BN, BD, NX, put A, O, C, respec- ^

tively, and let VZ be to XZ or DN as d to
bb

e, and VG be equal to

be equal to A — O, VG == ^=

then DN will

VZ =

O, and GD or NX — VZ — VG equal

274

term

THE MATHEMATICAL PRINCIPLES [BOOK H

nbb

nn -f- n

bb U ! J • x • £ • *°

•=rr be resolved into an infinite series -r- + —
A — Of A" A.n

3 -±- 3nn + 2/i

X O +

n

" ~ x bb O3,&c,,andGD will be equal

g^TT-T X 00 O2 +

c£ bb d nbb + ?m -
toC — -A--T-+-O- -r O - ~

e A" e A" + l 2An -f

+ H i^T 't"\ bb°3> &c- The second tcrm - O - -T

6An + e An 4- l

series is to be used for 0,0, the third ^ 0 66O2 for Roo, the fourth

— -£\~r~3 bbO5 for So3. And thence the density of the medium

Oof this

-, in any place G7 will be

2dnbb

nub*'

and therefore if in VZ you take VY equal to n X VG, that density is re-

n w IT- j ^ * 2rf//66 /mfi4

ciprocally as XY. For A2 and — A2 -- — A + — r are the

tc/ o^x ./\_ "

squares of XZ and ZY. Hut the resistance in the same place G is to the
force of gravity as 3S X - to 4RR, that is, as XY to

And the velocity there is the same wherewith the projected body would
move in a parabola, whose vertex is G, diameter GD, and latus rectum

2XY2

or - -------- - - . Q.E.I.

R

nn

VG

AC
HT

SCHOLIUM.

In the same manner that the den
sity of the medium comes out to be as

S X AC .

« Tjr™ m ^°r- 1) if the resistance

lx X HI

is put as any power V" of the velocity
V, the density of the medium will

come out to be as

B C D E Q

— . x

S

And therefore if a curve can be found, such that the ratio of to

4 — o

R— i-

SEC. II.J

OF NATURAL PHILOSOPHY,

275

n — 1

, or ofgr^ to

may be given ; the body, in an uni-

z

HT

AC

form medium, whose resistance is as the power V" of the velocity V, will
move in this curve. But let us return to more simple curves.

Because there can be no motion in a para
bola except in a non-resisting medium, but
in the hyperbolas here described it is produced
by a perpetual resistance ; it is evident that
the line which a projectile describes in an
uniformly resisting medium approaches nearer
to these hyperbolas than to a parabola. That
line is certainly of the hyperbolic kind, but
about the vertex it is more distant from the
asymptotes, and in the parts remote from the

vertex draws nearer to them than these hy- M JL BD K

perbolas here described. The difference, however, is not so great between
the one and the other but that these latter may be commod^ously enough
used in practice instead of the former. And perhaps these may prove more
useful than an hyperbola that is more accurate, and at the same time more
compounded. They may be made use of, then, in this manner.

Complete the parallelogram XYGT, and the right line GT will touch
the hyperbola in G, and therefore the density of the medium in G is re-

GT2

ciprocally as the tangent GT, and the velocity there as ^ -^=- ; and the

resistance is to the force of gravity as GT to

Therefore if a body projected from the
place A, in the direction of the right line
AH, describes the hyperbola AGK and
AH produced meets the asymptote NX in
H, arid AI drawrri parallel to it meets the
other asymptote MX in I ; the density of
the mediu.n in A will be reciprocally as
AH. and the velocity of the body as -J

AH1

. . . and the resis'an^e there to the force
Al

2nn

n +2

^- X GV.

. TT 2nn + 2n

of gravity rs AH to ZiTo ~

X AI. Her,ce the following rules a e

deduced.

RULE 1. If the density of the medium at A, and the velocity with which
the body is projected remain the same, and the angle NAH be changed ,
the lengths AH, AI, HX will remain. Therefore if those lengths, in any

276 THE MATHEMATICAL PRINCIPLES [BOOK II.

one case, are found, the hyperbola may afterwards be easily determined
from any given angle NAH.

RULE 2. If the angle NAH, and the density of the medium at A, re
main the same, and the velocity with which the body is projected be
changed, the length AH will continue the same ; and AI will be changed
in a duplicate ratio of the velocity reciprocally.

RULE 3. If the angle NAH, the velocity of the body at A, and the ac-
celerative gravity remain the same, and the proportion of the resistance at
A to the motive gravity be augmented in any ratio ; the proportion of AH
to A I will be augmented in the same ratio, the latus rectum of the above-

AH2

mentioned parabola remaining the same, and also the length propor-

AI

tional to it ; and therefore AH will be diminished in the same ratio, and
AI will be diminished in the duplicate of that ratio. But the proportion
of the resistance to the weight is augmented, when either the specific grav
ity is made less, the magnitude remaining equal, or when the density of
the medium is made greater, or when, by diminishing the magnitude, the
resistance becomes diminished in a less ratio than the weight.

RULE 4. Because the density of the medium is greater near the vertex
of the hyperbola than it is in the place A, that a mean density may be
preserved, the ratio of the least of the tangents GT to the tangent AH
ought to be found, and the density in A augmented in a ratio a little
greater than that of half the sum of those tangents to the least of the
tangents GT.

RULE 5. If the lengths AH, AI are given, and the figure AGK is to be
described, produce HN to X, so that HX may be to AI as n -\- 1 to 1 ; and
with the centre X, and the asymptotes MX, NX, describe an hyperbola
through the point A, such that AI may be to any of the lines VG as XV"

to xr.

RULE 6. By how much the greater the number n is, so much the more
accurate are these hyperbolas in the ascent of the body from A, and less
accurate in its descent to K ; and the contrary. The conic hyperbola
keeps a mean ratio between these, and is more simple than the rest. There
fore if the hyperbola be of this kind, and you are to find the point K,
where the projected body falls upon any right line AN passing through
the point A, let AN produced meet the asymptotes MX, NX in M and N,
and take NK equal to AM.

RULE 7. And hence appears an expeditious method of determining this
hyperbola from the phenomena. Let two similar and equal bodies be pro
jected with the same velocity, in different angles HAK, h A k, and let them
fall upon the plane of the horizon in K and k ; and note the proportion
of AK to A A". Let it be as d to e. Then erecting a perpendicular A I of
uny length, assume any how the length AH or Ah, and thence graphically,

SEC. II.

OF NATURAL PHILOSOPHY.

27?

or by scale and compass, collect the lengths AK, A/>* (by Rule 6). If the
ratio of AK to A/.* bo the same with that of d to e, the length of AH was

rightly assumed. If not, take on the indefinite right line SM, the length
SM equal to the assumed AH ; and erect a perpendicular MN equal to the

AK d

difference -r-r of the ratios drawn into any given right line. By the

like method, from several assumed lengths AH, you may find several points
N ; and draw througli them all a regular curve NNXN, cutting tr.e right
line SMMM in X. Lastly, assume AH equal to the abscissa SX, and
thence find again the length AK ; and the lengths, w'hich are to the as
sumed length AI, and this last AH, as the length AK known by experi
ment, to the length AK last found, will be the true lengths AI and AH,
which were to be found. But these being given, there will be given also
the resisting force of the medium in the place A, it being to the force of
gravity as AH to JAI. Let the density of the medium be increased by
Rule 4, and if the resisting force just found be increased in the same ratio,
it will become still more accurate.

RULE 8. The lengths AH, HX being found ; let there be now re
quired the position of the line AH, according to which a projectile thrown
with that given velocity shall fall upon any point K. At the joints A
and K, erect the lines AC, KF perpendicular to the horizon : whereof let
AC be drawn downwards, and be equal to AI or ^HX. With the asymp
totes AK, KF, describe an hyperbola, whose conjugate shall pass through
the point C ; and from the centre A, with the interval AH. describe a cir
cle cutting that hyperbola in the point H ; then the projectile thrown in
the direction of the right line AH will fall upon the point K. Q.E.I. For
the point H, because of the given length AH, must be somewhere in the
circumference of the described circle. Draw CH meeting AK and KF in
E and F: and because CH, MX are parallel, and AC, AI equal, AE will
be equal to AM, and therefore also equal to KN. But CE is to AE as
FH to KN. and therefore CE and FH are equal. Therefore the point H
falls upon the hyperbolic curve described with the asymptotes AK,.KF
whose conjugate passes through the point C ; and is therefore found in the

27S

THE MATHEMATICAL PRINCIPLES

[BOOK 11

common intersection of this hyperbolic
curve and the circumference of the de
scribed circle. Q.E.D. It is to be ob
served that this operation is the same,
whether the right line AKN be parallel to
the horizon, or inclined thereto in any an
gle : and that from two intersections H,
//., there arise two angles NAH, NAA ;
and that in mechanical practice it is suf
ficient once to describe a circle, then to
apply a ruler CH, of an indeterminate length, HO to the point C, that its
part PH, intercepted between the circle and the right line FK, may bo
equal to its part CE placed between the point C and the right line AK

What has been said of hyperbolas may be easily
applied to pir i'>;>l.i3. For if a parabola be re
presented by XAGK, touched by a right line XV
in the vertex X, and the ordinates IA, YG be as
any powers XI", XV"; of the abscissas XI, XV ;
draw XT, GT, AH, whereof let XT be parallel
to VG, and let GT, AH touch the parabola in
G and A : and a body projected from any place
A, in the direction of the right line AH, with a
due velocity, will describe this parabola, if the density of the medium in
each of the places G be reciprocally as the tangent GT. In that case the
velocity in G will be the same as would cause a body, moving in a non-
resisting space, to describe a conic parabola, having G for its vertex, VG

2GT2

produced downwards for its diameter, and -. — for its latus

nn — n X VG

rectum. And the resisting force in G will be to the force of gravity as GT to

2nti — 2tt

~2~ VG. Therefore if NAK represent an horizontal line, and botli

the density of the medium at A, and the velocity with which the body is
projected, remaining the same, the angle NAH be any how altered, the
lengths AH, AI, HX will remain; and thence will be given the vertex X
of the parabola, and the position of the right line XI ; and by taking VG
to IA as XVn to XI", there will be given all the points G of the parabola,
through which the projectile will pass.

SEC. IILJ OF NATURAL PHILOSOPHY. 279

SECTION III.

Of the motions of bodies which are resisted partly in the ratio of the ve
locities, and partly in the duplicate of the same ratio.

PROPOSITION XI. THEOREM VIII.

If a body be resisted partly in the ratio and partly in the duplicate ratio
of its velocity, and moves in a similar medium by its innate force
only; and the times be taken in arithmetical progression; then
quantities reciprocally proportional to the velocities, increased by a cer
tain given quantity, will be in geometrical progression.
With the centre C, and the rectangular asymptotes ^

OADd and CH, describe an hyperbola BEe, and let | \p

AB, DE, de. be parallel to the asymptote CH. In |

the asymptote CD let A, G be given points ; and if

the time be expounded by the hyperbolic area ABED

uniformly increasing, I say, that the velocity may ~r

be expressed by the length DF, whose reciprocal

GD, together with the given line CG, compose the

length CD increasing in a geometrical progression.

For let the areola DEec/ be the least given increment of the time, and

Dd will be reciprocally as DE, and therefore directly as CD. Therefore

the decrement of ^TR, which (by Lem. II, Book II) is ^no, will be also as

D
tf

CD CG + GD 1 CG

GO* °r — GD2 ~ ' fc 1S>aS GD + GJD2* *nerefore tne timc
uniformly increasing by the addition of the given particles EDcfe, it fol

lows that r decreases in the same ratio with the velocity. For the de

crement of the velocity is as the resistance, that is (by the supposition), as
the sum of two quantities, whereof one is as the velocity, and the other as

the square of the velocity ; and the decrement of ~~ is as the sum of the

1 C^(^ 1

quantities ~-^=r and pfp,> whereof the first is ^^r itself, and the last

i i

is a* /-TFT; • therefore T^-R is as tne velocity, the decrements of both

-' CilJ

being analogous. And if the quantity GD reciprocally proportional to
T, be augmented by the given quantity CG ; the sum CD, the time

ABED uniformly increasing, will increase !n a geometrical progression.
Q.E.D.

THE MATHEMATICAL PRINCIPLES [BOOK II

COR. 1. Therefore, if, having the points A and G given, the time bo
expounded by the hyperbolic area ABED, the velocity may be expounded

by -r the reciprocal of GD.

COR. 2. And by taking GA to GD as the reciprocal of the velocity at
the beginning to the reciprocal of the velocity at the end of any time
ABED, the point G will be found. And that point being found the ve
locity may be found from any other time given.

PROPOSITION XII. THEOREM IX.

The same things being supposed, I say, that if the spaces described are
taken in arithmetical progression, the velocities augmented by a cer
tain given quantity will be in geometrical progression.

In the asymptote CD let there be given the
point R, and, erecting the perpendicular RS
meeting the hyperbola in S, let the space de
scribed be expounded by the hyperbolic area
I RSED ; and the velocity will be as the length

J GD, which, together with the given line CG,

** composes a length CD decreasing in a geo
metrical progression, while the space RSED increases in an arithmetical
[(regression.

For, because the incre nent EDde of the space is given, the lineola DC?,
which is the decrement of GD, will be reciprocally as ED, and therefore
directly as CD ; that is, as the sum of the same GD and the given length
CG. But the decrement of the velocity, in a time reciprocally propor
tional thereto, in which the given particle of space D^/eE is described, is
as the resistance and the time conjunctly, that is. directly as the sum of
two quantities, whereof one is as the velocity, the other as the square of
the velocity, and inversely as the veh city ; and therefore directly as the
sum of two quantities, one of which is given, the other is- as the velocity.
Therefore the decrement both of the velocity and the line GD is as a given
quantity and a decreasing quantity conjunctly; and, because the decre
ments are analogous, the decreasing quantities will always be analogous ;
viz., the velocity, and the line GD. U.E.D.

COR. 1. If the velocity be expounded by the length GD, the space de
scribed will be as the hyperbolic area DESR.

COR. 2. And if the point „ be assumed any how, the point G will be
found, by taking GR to GD as the velocity at the beginning to the velo
city after any space RSED is described. The point G being given, the
space is given from the given velocity : and the contrary.

Cotw 3. Whence since (by Prop. XI) the velocity is given from the given

SEC. Ilt.1

Or NATURAL PHILOSOPHY.

281

time, and (by this Prop.) the space is given from the given velocity ; the
space will be given from the given time : and the contrary.

PROPOSITION XKI. THEOREM X.

Supposing that a body attracted downwards by an uniform gravity as
cends or descends in a right line; and that the same is resisted
partly in the ratio of its velocity, and partly in the duplicate ratio
thereof: I say, that, if right lines parallel to the diameters of a circle
and an hyperbola, be drawn through the ends of the, conjugate diame
ters, and the velocities be as some segments of those parallels drawn
from a given point, the times will be as the sectors of the, areas cut
off by right lines drawn from the centre to the ends of the segments ;
and the contrary.

CASE 1. Suppose first that the body is ascending,
and from the centre D, with any semi-diameter DB,
describe a quadrant BETF of a circle, and through
the end B of the semi-diameter DB draw the indefi
nite line BAP, parallel to the semi-diameter DF. In
chat line let there be given the point A, and take the
segment AP proportional to the velocity. And since
one part of the resistance is as the velocity, and
another part as the square of the velocity, let the
whole resistance be as AP2 -f 2BAP. Join DA, DP, cutting the circle
in E and T, and let the gravity be expounded by DA2, so that the gravity
shall be to the resistance in P as DA2 to AP2+2BAP ; and the time of the
whole ascent will be as the sector EDT of the circle.

For draw DVQ,, cutting off the moment PQ, of the velocity AP, and the
moment DTV of the sector DET answering to a given moment of time ;
and that decrement PQ, of the velocity will be as the sum of the forces of
gravity DA2 and of resistance AP2 + 2BAP, that is (by Prop. XII
BookII,Elem.),asDP*. Then the arsa DPQ, which is proportional to PQ:
is as DP2, and the area DTV, which is to the area DPQ, as DT2 to DP2, it
as the given quantity DT2. Therefore the area EDT decreases uniformly
according to the rate of the future time, by subduction of given particles DT V,
and is therefore proportional to the time of the whole ascent. Q..E.D.

CASE 2. If the velocity in the ascent
of the body be expounded by the length
AP as before, and the resistance be made
as AP2 -f- 2BAP,and if the force of grav
ity be less than can be expressed by DA2 ;
take BD of such a length, that AB2 —
BD2 maybe proportional to the gravity,
and let DF be perpendicular and equal

F O

£S2 THE MATHEMATICAL PRINCIPLES [BOOK ll.

to DB, and through the vertex F describe the hyperbola FTVE, whose con
jugate semi -diameters are DB and DF; and which cuts DA in E, and DP,
DQ in T and V ; and the time of the whole ascent will be as the hyper
bolic sector TDE.

For the decrement PQ of the velocity, produced in a given particle of
time, is as the sum of the resistance AP2 -f 2BAP and of the gravity
AB2 — BD2, that is, as BP2 — BD2. But the area DTV is to the area
DPQ as DT2 to DP2 ; and, therefore, if GT be drawn perpendicular to
DF. as GT2 or GD2 — DF2 to BD2, and as GD2 to BP2, and, by di
vision, as DF2 to BP2 — BD2. Therefore since the area DPQ is as PQ,
that is, as BP2 — BD2, the area DTV will be as the given quantity DF2.
Therefore the area EDT decreases uniformly in each of the equal particles
of time, by the subduction of so many given particles DTV, and therefore
is proportional to the time. Q.E.D.

CASE 3. Let AP be the velocity in the descent of
""" the body, and AP2 + 2BAP the force of resistance,
and BD2 — AB2 the force of gravity, the angle DBA
being a right one. And if with the centre D, and the
principal vertex B, there be described a rectangular
hyperbola BETV cutting DA, DP, and DQ produced
in E, T, and V : the sector DET of this hyperbola will
D be as the whole time of descent.

For the increment PQ of the velocity, and the area DPQ proportional
to it, is as the excess of the gravity above the resistance, that is, as
m)2?_ AB2 _2BAP — AP2 or BD2— BP2. And the area DTV
is to the area DPQ as DT3 to DP2 ; and therefore as GT2 or GD" -
BD2 to BP2, and as GD2 to BD2, and, by division, as BD2 to BD2 -
BP2. Therefore since the ami DPQ is as BD2 — BP2, the area DTV
will be as the given quantity BD2. Therefore the area EDT increases
uniformly in the several equal particles of time by the addition of as
many given particles DTV, and therefore is proportional to the time of
the descent. Q.E.D.

Con. If with the centre D and the semi-diameter DA there be drawn
through the vertex A an arc A/ similar to the arc ET, and similarly sub-
tendino^the angle A DT, the velocity AP will be to the velocity which the
body in the time EDT, in a non-resisting space, can lose in its ascent, or
acquire in its descent, as the area of the triangle DAP to the area of the
Bector DA/ ; and therefore is given from the time given. For the velocity
ir a non-resistin^ medium is proportional to the time, and therefore to this
sector : in a resisting medium, it is as the triangle ; and in both mediums,
where it is least, it approaches to the ratio of equality, as the sector and
triangle do

SEC. III.]

OF NATURAL PHILOSOPHY.

283

SCHOLIUM.

One may demonstrate also that case in the ascent of the body, where the
force of gravity is less than can be expressed by DA2 or AB2 + BD2, and
greater than can be expressed by AB2 — DB2, and must be expressed by
AB2. But I hasten to other things

PROPOSITION XIV. THEOREM XL

The same things being supposed, 1 say, that the space described in the
ascent or descent is as the difference of the area by which the time is
expressed, and of some other area which is augmented or diminished
in an arithmetical progression ; if the forces compounded of the re
sistance and the gravity be taken, in a geometrical progression.
Take AC (in these three figures) proportional to the gravity, and AK

to the resistance ; but take them on the same side of the point A, if the

\*

"1

\

B A

K QP

body is descending, otherwise on the contrary. Erect A b, which make to
DB as DB2 to 4BAC : and to the rectangular asymptotes CK, CH, de
scribe the hyperbola 6N ; and, erecting KN perpendicular to CK, the area
A/AK will be augmented or diminished in an arithmetical progression,
while the forces CK are taken in a geometrical progression. I say, there
fore, that the distance of the body from its greatest altitude is as the excess
of the area A6NK above the area DET.

For since AK is as the resistance, that is, as AP2 X 2BAP ; assume

any given quantity Z, and put AK equal to

then (by Lem,

284

THE MATHEMATICAL PRINCIPLES [BOOK II

2APQ, + 2B A X PU

II of this Book) the moment KL of AK will be equal to
2BPQ

or

Z

-, and the moment KLON of the area A£NK will be equal to

2BPQ X LO BPQ, X BD3
~~Z~~ >r 2Z X OK x~AB"

CASE 1. Now if the body ascends, and the gravity be as AB2 + BD9

BET being a circle, the line AC, which is proportional to the gravity

AW2 i RF)2

will be - ~ -- , and DP2 or AP2 + 2BAP + AB2 + BD2 will be

AK XZ + AC X Z or CK X Z : and therefore the area DTV will be to
the area DPQ, as DT2 or I)B2 to CK X Z.

CASE 2. If the body ascends, and the gravity be as AB2 — BD2, the
A r>2 _ HI)2

line AC will be " -- ^ --- , and DT2 will be to DP2 as DF2 or DB2
Z

to BP2— BD2 or AP2 + 2BAP + AB2 — BD2, that is, to AK X Z +

H

AC X Z or CK X Z. And therefore the area DTV will be to the area
DPQ as DB2 to CK X Z.

CASE 3. And by the same reasoning, if the body descends, and therefore
the gravity is as BD2 — AB2, and the line AC becomes equal to

or)2 AB2

5T— r ; the area DTV will be to the area DPQ as DB2 to CK X

Z

Z : as above.

Since, therefore, these areas are always in this ratio, if for the area

SEC. 111^ OF NATURAL PHILOSOPHY. 285

DTY, by which the moment of the time, always equal to itself, is express
ed, there be put any determinate rectangle, as BD X m, the area DPQ,,
that is, |BD X PQ, will be to BD X mas CK X Z to BD2. And thence
PQ X BD3 becomes equal to 2 BD XmX CK X Z, and the moment KLON

BP X BD X tn
of the area A6NK, found before, becomes - .-^ — -- . From the area

DET subduct its moment DTV or BD X m, and there will remain
--- -Pp — — . Therefore the difference of the moments, that is, the

AP X BD X m

mo.nent of the difference of tne areas, is equal to -- 7-5 --- ; and

therefore (because of the given quantity --- T-~ — ) as the velocity AP ;

that is, as the moment of the space which the body describes in its ascent
or descent. And therefore the difference of the areas, and that space, in
creasing or decreasing by proportional moments, and beginning together or
vanishing together, are proportional. Q.E.D.

COR. If the length, which arises by applying the area DET to the line
BD, be called M ; and another length V be taken in that ratio to the length
M, which the line DA has to the line DE; the space which a body, in a
resisting medium, describes in its whole ascent or descent, will be to the
space which a body, in a non-resisting medium, falling from rest, can de
scribe in the same time, as the difference of the aforesaid areas to

BD X V2
- -TO"""" j an(^ therefore is given from the time given. For the space in a

A.LJ

non-resisting medium is in a duplicate ratio of the time, or as V2 ; and.

BD X V2
because BD and AB are given, as --- -TTT- — . This area is equal to the

DA2 X BD x M2
area -- fvGr*~~~T~R — "~ anc* ^ne moment Of M is m ; and therefore the

DA2 X BD X 2M X m

moment of this area is --- =- --- ^5 — -. But this moment is to

"" X .A D

the moment of the difference of the aforesaid areas DET and A6NK, viz., to

AP X BD X m DA2 X BD X Mx DA2 . ^^

- -- , as - -r- - to |BD X AP, or as into DET

to DAP ; and, therefore, when the areas DKT and DAP are least, in the

BD X V2
ratio of equality. Therefore the area — r^ -- and the difference of the

areas DET and A&NK, when all these areas are least, have equal moments ;
and { re therefore equal. Therefore since the velocities, and therefore also
the s] aces in both mediums described together, in the beginning of the de
scent or the end of the ascent, approach to equality, and therefore are then

286 THE MATHEMATICAL PRINCIPLES [BOOK II

BD X V2

one to another as the area r-^ — , and the difference of the areas DET

AD

and A6NK ; and moreover since the space, in a non-resisting medium, is

BD X V2

perpetually as Tu~~> an(^ tne sPace> in a resisting medium, is perpetu
ally as the difference of the areas DET and A&NK ; it necessarily follows,
that the spaces, in both mediums, described in any equal times, are one to

BD X V2
another as that area 7-5 — » an(^ ^he difference of the areas DET and

A6NK. Q.E.D.

SCHOLIUM.

The resistance of spherical bodies in fluids arises partly from the tena
city, partly from the attrition, and partly from the density of the medium.
And that part of the resistance which arises from the density of the fluid
is, as I said, in a duplicate ratio of the velocity ; the other part, which
arises from the tenacity of the fluid, is uniform, or as the moment of the
time ; and, therefore, we might now proceed to the motion of bodies, whicli
are resisted partly by an uniform force, or in the ratio of the moments of
the time, and partly in the duplicate ratio of the velocity. But it is suf
ficient to have cleared the way to this speculation in Prop. VIII and IX
foregoing, and their Corollaries. For in those Propositions, instead of the
uniform resistance made to an ascending body arising from its gravity,
one may substitute the uniform resistance which arises from the tenacity
of the medium, when the body moves by its vis insita alone ; and when the
body ascends in a right line, add this uniform resistance to the force of
gravity, and subduct it when the body descends in a right line. One
might also go on to the motion of bodies which are resisted in part uni
formly, in part in the ratio of the velocity, and in part in the duplicate
ratio of the same velocity. And I have opened a way to this in Prop.
XIII and XIV foregoing, in which the uniform resistance arising from the
tenacity of the medium may be substituted for the force of gravity, or be
compounded with it as before. But I hasten to other things.

SKC. -IV'.] OF NATUEAL PHILOSOPHY. 2S?

SECTION IV.

Of the circular motion of bodies in resisting' mediums.

LEMMA III.

Let PQR be a spiral rutting all the radii SP, SO, SR, <J*c., in equal
angles. Draw tfie right line PT touching the spiral in any point P,
and cutting the radius SQ in T ; cfo'er?0 PO, QO perpendicular to
the spiral, and meeting- in, O, and join SO. .1 say, that if Hie points
P a/*(/ Q approach and coincide, the angle PSO vri/Z become a right
angle, and the ultimate ratio of the rectangle TQ, X 2PS to P^3 //>i//
/>e £/ie ya/io o/" equality.
For from the right angles OPQ, OQR, sub
duct the equal angles SPQ, SQR, and there
will remain the equal angles OPS, OQS.
Therefore a circle which passes through the
points OSP will pass also through the point
Q. Let the points P and Q, coincide, and
this circle will touch the spiral in the place
of coincidence PQ, and will therefore cut the
right line OP perpendicularly. Therefore OP will become a diameter of
this circle, and the angle OSP, being in a semi-circle, becomes a right
one. Q.E.1).

Draw Q,D, SE perpendicular to OP, and the ultimate ratios of the lines
will be as follows : TO to PD as TS or PS to PE, or 2PO to 2PS • and
PD to PO as PO to 2PO ; and, ex cequo pertorbatt, to TO to PO as PO
to 2PS. Whence PO2 becomes equal to TO X 2PS. O.E.D.

PROPOSITION XV. THEOREM XII.

Tf the density of a medium in each place thereof be recipr on iJl y as the
distance of the places from an immovable centre, aud the centripetal
force be in the duplicate ratio of the density ; I say, that a body mny
revolve in a spiral which cuts all the radii drawn from that centre
in a given angle.

Suppose every thing to be as in the forego
ing Lemma, and produce SO to V so that SV
may be equal to SP. In any time let a body,
in a resisting medium, describe the least arc
PO, and in double the time the least arc PR :
and the decrements of those arcs arising from
the resistance, or their differences from the
arcs which would be described in a non-resist
ing medium in the same times, will be to each
other as the squares of the times in which they
are generated ; therefore the decrement of the

288 THE MATHEMATICAL PRINCIPLES [_BoOK 11

arc PQ is the fourth part of the decrement of the arc PR. Whence also
if the area QSr be taken equal to the area PSQ, the decrement of the arc
PQ will be equal to half the lineola Rr ; and therefore the force of resist
ance and the centripetal force are to each other as the lineola jRrandTQ
which they generate in the same time. Because the centripetal force with
which the body is urged in P is reciprocally as SP2, and (by Lem. X,
Book I) the lineola TQ, which is generated by that force, is in a ratio
compounded of the ratio of this force and the duplicate ratio of the time
in which the arc PQ, is described (for in this case I neglect the resistance,
as being infinitely less than the centripetal force), it follows that TQ X
SP2, that is (by the last Lemma), fPQ2 X SP, will be in a duplicate ra
tio of the time, and therefore the time is as PQ, X v/SP ; and the velo
city of the body, with which the arc PQ is described in that time, as

PQ 1

— -«p or , that is, in the subduplicate ratio of SP reciprocally.

And, by a like reasoning, the velocity with which the arc QR is described,
is in the subduplicate ratio of SQ reciprocally. Now those arcs PQ and
QR are as the describing velocities to each other ; that is, in the subdu

plicate ratio of SQ to SP, or as SQ to x/SP X SQ; and, because of the
equal angles SPQ, SQ?', and the equal areas PSQ, QSr, the arc PQ is to
the arc Qr as SQ to SP. Take the differences of the proportional conse
quents, and the arc PQ will be to the arc Rr as SQ to SP — VSP X ~SQ~,
or ^VQ. For the points P and Q coinciding, the ultimate ratio of SP —

X SQ to |VQ is the ratio of equality. Because the decrement of
the arc PQ arising from the resistance, or its double Rr, is as the resistance

and the square of the time conjunctly, the resistance will be &Sp-^r — op.

*1

X
But PQ was to Rr as SQ to |VQ, and thence SSaTXToD becomes as

Jr vst X oJr

-VQ -OS

pWxsvxSQ,' or ns ETp^TsP- For the poillts p and a coincidin&

SP and SQ coincide also, and the angle PVQ becomes a right one; and,
because of the similar triangles PVQ, PSO, PQ. becomes to '-VQ as OP

OS

to | OS. Therefore : y — — -- is as the resistance, that is, in the ratio of
\J i X ol

the density of the medium in P and the duplicate ratio of the velocity
conjunc-tly. Subduct the duplicate ratio of the velocity, namely, the ratio

1 OS

^5, and there will remain the density of the medium in P. as 7^5 - =

OA Ur X fei

Let the spiral be given, and; because of the given ratio of OS to OP, the
density of the medium in P will be as ~-p. Therefore in a medium whose

SEC. IV,] OF NATURAL PHILOSOPHY. 2S9

density is reciprocally as SP the distance from the centre, a body will re
volve in this spiral. Q.E.D.

COR. 1. The velocity in any place P, is always the same wherewith a
body in a non-resisting medium with the same centripetal force would re
volve in a circle, at the same distance SP from the centre.

COR. 2. The density of the medium, if the distance SP be given, is as

OS OS

TTp, but if that distance is not given, as ^ ^5. And thence a spiral

may be fitted to any density of the medium.

COR. 3. The force of the resistance in any place P is to the centripetal
force in the same place as |OS to OP. For those forces are to each other

^VQ x PQ iPQ2

as iRr and TQ, or as 1—^-^~- and ^-, that is, as iVQ and PQ,

ol%, ol

or |OS and OP. The spiral therefore being given, there is given the pro
portion of the resistance to the centripetal force ; and, vice versa, from that
proportion given the spiral is given.

COR. 4. Therefore the body cannot revolve in this spiral, except where
the force of resistance is less than half the centripetal force. Let the re
sistance be made equal to half the centripetal force, and the spiral will co
incide with the right line PS, and in that right line the body will descend
to the centre with a velocity that is to the velocity, with which it was
proved before, in the case of the parabola (Theor. X, Book I), the descent
would be made in a non-resisting medium, in the subduplicate ratio of
unity to the number two. And the times of the descent will be here recip
rocally as the velocities, and therefore given.

COR. 5. And because at equal distances
from the centre the velocity is the same in the
spiral PQ,R as it is in the right line SP, and
the length of the spiral is to the length of the
right line PS in a given ratio, namely, in the
ratio of OP to OS ; the time of the descent in
the spiral will be to the time of the descent in
the right line SP in the same given ratio, and
therefore given.

COR. 6. If from the centre S, with any two
given intervals, two circles are described ; and
these circles remaining, the angle which the spiral makes with the radius"
PS be any how changed ; the number of revolutions which the body can
complete in the space between the circumferences of those circles, going

PS

round in the spiral from one circumference to another, will be as 7^, or as

Ok5'

ths tangent of the angle which the spiral makes with the radius PS ; and

19

290 THE MATHEMATICAL PRINCIPLES [BOOK II

OP

the time of the same revolutions will be as -^, that is, as the secant of the

Uo

same angle, or reciprocally as the density of the medium.

COR. 7. If a body, in a medium whose density is reciprocally as the dis
tances of places from the centre, revolves in any curve AEB about that
centre, and cuts the first radius AS in the same
angle in B as it did before in A, and that with a
velocity that shall be to its first velocity in A re
ciprocally in a subduplicate ratio of the distances
from the centre (that is, as AS to a mean propor
tional between AS and BS) that body will con
tinue to describe innumerable similar revolution?
BFC, CGD, &c., and by its intersections will
distinguish the radius AS into parts AS, BS, CS, DS, &c., that are con
tinually proportional. But the times of the revolutions will be as the
perimeters of the orbits AEB, BFC, CGD, &c., directly, and the velocities

3 3

at the beginnings A, B, C of those orbits inversely ; that is as AS % BS %

CS"2". And the whole time in which the body will arrive at the centre,
will be to the time of the first revolution as the sum of all the continued

142

proportionals AS2, BS2, CS2, going on ad itifinitum, to the first term

* i 3

AS2 ; that is, as the first term AS2 to the difference of the two first AS2

— BS% or as f AS to AB very nearly. Whence the whole time may be
easily found.

COR. 8. From hence also may be deduced, near enough, the motions of
bodies in mediums whose density is either uniform, or observes any other
assigned law. From the centre S, with intervals SA, SB, SC, &c., con
tinually proportional, describe as many circles ; and suppose the time of
the revolutions between the perimeters of any two of those circles, in the
medium whereof we treated, to be to the time of the revolutions between
the same in the medium proposed as the mean density of the proposed me
dium between those circles to the mean density of the medium whereof we
treated, between the same circles, nearly : and that the secant of the angle
in which the spiral above determined, in the medium whereof we treated,
cuts the radius AS, is in the same ratio to the secant of the angle in which
the new spiral, in the proposed medium, cuts the same radius : and also
that the number of all the revolutions between the same two circles is nearly
as the tangents of those angles. If this be done every where between every
two circles, the motion will be continued through all the circles. And by
this means one may without difficulty conceive at what rate and in what
time bodies ought to revolve in any regular medium.

•SEC. IY.1 OF NATURAL PHILOSOPHY. 291

COR. 9. And although these motions becoming eccentrical should be
performed in spirals approaching to an oval figure, yet, conceiving the
several revolutions of those spirals to be at the same distances from each
other, and to approach to the centre by the same degrees as the spiral above
described, we may also understand how the motions of bodies may be per
formed in spirals of that kind.

PROPOSITION XVI. THEOREM XIII.

If the density of the medium in each of the places be reciprocally as the
distance of the >, places from the immoveable centre, and the centripetal
force be reciprocally as any power of the same distance, I say, that the
body may revolve in a spiral intersecting all the radii drawn from
that centre in a given, angle.
This is demonstrated in the same manner as

the foregoing Proposition. For if the centri

petal force in P be reciprocally as any power

SPn+1 of the distance SP whose index is n

+ 1 ; it will be collected, as above, that the

time in which the body describes any arc PQ,

i
will be as PQ, X PS2U ; and the resistance in

i!! x _

n;°raS "~'

X SPn;PQ, X SP"XSQ,'

, , 1 — in X OS . 1 — \n X OS .

therefore as Qp"^~gpirqTT' tliat 1S> (because - ~~Qp~~ 1S a £lven

quantity), reciprocally as SPn+ ! . And therefore, since the velocity is recip
rocally as SP3", the density in P will be reciprocally as SP.

COR. 1. The resistance is to the centripetal force as 1 — ^//. X OS
to OP.

COR. 2. If the centripetal force be reciprocally as SP3. 1 — ±w will be
=== 0 ; and therefore the resistance and density of the medium will be
nothing, as in Prop. IX, Book I.

COR. 3. If the centripetal force be reciprocally as any power of the ra
dius SP, whose index is greater than the number 3, the affirmative resist
ance will be changed into a negative.

SCHOLIUM.

This Proposition and the former, which relate to mediums of unequal
density, are to be understood of the motion of bodies that are so small, that
the greater density of the medium on one side of the body above that on
the other is not to be considered. I suppose also the resistance, cateris
paribus, to be proportional to its density. Whence, in mediums whose

292 THE MATHEMATICAL PRINCIPLES | BoOK II

force of resistance is not as the density, the density must be so much aug
mented or diminished, that either the excess of the resistance may be taken
away, or the defect supplied.

PROPOSITION XVII. PROBLEM IV

To find the centripetal force and the resisting force of the medium, by
which a body, the law of the velocity being given, shall revolve in a
given spiral.

Let that spiral be PQR. From the velocity,
with which the body goes over the very small arc
PQ,, the time will be given : and from the altitude
TQ,, which is as the centripetal force, and the
square of the time, that force will be given. Then
from the difference RSr of the areas PSQ, and
Q,SR described in equal particles of time, the re
tardation of the body will be given ; and from
the retardation will be found the resisting force
and density of the medium.

PROPOSITION XVIII. PROBLEM V.

The law of centripptal force being given, to find the density of the me
dium in each of the places thereof, by which a body may describe a
given spiral.

From the centripetal force the velocity in each place must be found ;
then from the retardation of the velocity the density of the medium is
found, as in the foregoing Proposition.

But I have explained the method of managing these Problems in the
tenth Proposition and second Lemma of this Book; and will no longer
detain the reader in these perplexed disquisitions. I shall now add some
things relating to the forces of progressive bodies, and to the density and
resistance of those mediums in which the motions hitherto treated of, and
those akin to them, are performed.

SEC. V.] OF NATURAL PHILOSOPHY. 293

SECTION V.

Of the density and compression of fluids ; and of hydrostatics.

THE DEFINITION OF A FLUID.

A fluid is any body whose parts yield to any force impressed on it,
by yielding, are easily moved among themselves.

PROPOSITION XIX. THEOREM XIv

All the parts of a homogeneous and unmoved fluid included in any un
moved vessel, and compressed on every side (setting aside the consider
ation of condensation, gravity, and all centripetal forces], will be
equally pressed on every side, and remain in their places without any
motion arising from that pressure.
CASE 1. Let a fluid be included in the spherical
vessel ABC, and uniformly compressed on every
side : 1 say, that no part of it will be moved by
that pressure. For if any part, as L), be moved,
all such parts at the same distance from the centre
on every side must necessarily be moved at the
same time by a like motion ; because the pressure
of them all is similar and equal ; and all other
motion is excluded that does not arise from that
pressure. But if these parts come all of them nearer to the centre, the
fluid must be condensed towards the centre, contrary to the supposition.
If they recede from it, the fluid must be condensed towards the circumfer
ence ; which is also contrary to the supposition. Neither can they move
in any one direction retaining their distance from the centre, because for
the same reason, they may move in a contrary direction : but the sami
part cannot be moved contrary ways at the same time. Therefore no
part of the fluid will be moved from its place. Q,.E.T).

CASE 2. I say now, that all the spherical parts of this fluid are equally
pressed on every side. For let EF be a spherical part of the fluid ; if this
be not pressed equally on every side, augment the lesser pressure till it be
pressed equally on every side; and its parts (by Case I) will remain in
their places. But before the increase of the pressure, they would remain
in their places (by Case 1) ; and by the addition of a new pressure they
will be moved, by the definition of a fluid, from those places. Now these
two conclusions contradict each other. Therefore it was false to say that
the sphere EF was not pressed equally on every side. Q,.E.D.

CASE 3. I say besides, that different spherical parts have equal pressures.
For the contiguous spherical parts press each other mutually and equally
in the point of contact (by Law III). But (by Case 2) they are pressed on
every side with the same force. Therefore any two spherical parts lot

391 THE MATHEMATICAL PRINCIPLES [BoOK II.

contiguous, since an intermediate spherical part can touch both, will be
pressed with the same force. Q.E.D.

CASE 4. I say now, that all the parts of the fluid are every where press
ed equally. For any two parts may be touched by spherical parts in any
points whatever ; and there they will equally .press those spherical parts
(by Case 3). and are reciprocally equally pressed by them (by Law III).
Q.E.D.

CASE 5. Since, therefore, any part GHI of the fluid is inclosed by the
rest of the fluid as in a vessel, and is equally pressed on every side ; and
also its parts equally press one another, and are at rest among themselves ;
it is manifest that all the parts of any fluid as GHI, which is pressed
equally on every side, do press each other mutually and equally, and are at
rest among themselves. Q.E.D.

CASE 6. Therefore if that fluid be included in a vessel of a yielding
substance, or that is not rigid, and be not equally pressed on every side,
the same will give way to a stronger pressure, by the Definition of fluidity.

CASE 7. And therefore, in an inflexible or rigid vessel, a fluid will not
Sustain a stronger pressure on one side than on the other, but will give
way to it, and that in a moment of time ; because the rigid side of the
vessel does not follow the yielding liquor. But the fluid, by thus yielding,
will press against the opposite side, and so the pressure will tend on every
side to equality. And because the fluid, as soon as it endeavours to recede
from the part that is most pressed, is withstood by the resistance of the
vessel on the opposite side, the pressure will on every side be reduced to
equality, in a moment of time, without any local motion : and from thence
the parts of the fluid (by Case 5) will press each other mutually and equal
ly, and be at rest among themselves. Q..E.D.

COR. Whence neither will a motion of the parts of the fluid among
themselves be changed by a pressure communicated to the external super
ficies, except so far as either the figure of the superficies may be somewhere
altered, or that all the parts of the fluid, by pressing one another more in
tensely or remissly, may slide with more or less difficulty among them
selves.

PROPOSITION XX. THEOREM XV.

Jf all the parts of a spherical fluid, homogeneous at equal distances from
the centre, lying on. a spherical concentric bottom, gravitate towards
the centre of the whole, the bottom will sustain the weight of a cylin
der, whose base is equal to the superficies of the bottom, and whose al
titude is the same with that of the incumbent fluid.
Let OHM be the superficies of the bottom, and AEI the upper super
ficies of the fluid. Let the fluid be distinguished into concentric orbs of
3qual thickness, by the innumerable spherical superficies *3PK, CGL : and

SEC. V OF NATURAL PHILOSOPHY. 295

conceive the force of gravity to act only in the
upper superficies of every orb, and the actions
to be equal on the equal parts of all the su
perficies. Therefore the upper superficies AE
is pressed by the single force of its own grav
ity, by which all the parts of the upper orb,
and the second superficies BFK, will (by
Prop. XIX), according to its measure, be
equally pressed. The second superficies BFK
is pressed likewise by the force of its own
gravity, which, added to the former force,
makes the pressure double. The third superficies CGL is, according to its
measure, acted on by this pressure and the force of its own gravity besides,
which makes its pressure triple. And in like manner the fourth superfi
cies receives a quadruple pressure, the fifth superficies a quintuple, and so
on. Therefore the pressure acting on every superficies is not as the solid
quantity of the incumbent fluid, but as the number of the orbs reaching
to the upper surface of the fluid ; and is equal to the gravity of the lowest
orb multiplied by the number of orbs : that is, to the gravity of a solid
whose ultimate ratio to the cylinder above-mentioned (when the number of
the orbs is increased and their thickness diminished, ad infiititum, so that
the action of gravity from the lowest superficies to the uppermost may be-
some continued) is the ratio of equality. Therefore the lowest superficies
•sustains the weight of the cylinder above determined. Q..E.D. And by a
like reasoning the Proposition will be evident, where the gravity of the
fluid decreases in any assigned ratio of the distance from the centre, and
also where the fluid is more rare above and denser below. Q.E.D.

COR. 1. Therefore the bottom is not pressed by the whole weight of the
incumbent fluid, but only sustains that part of it which is described in the
Proposition ; the rest of the weight being sustained archwise by the spheri
cal figure of the fluid.

COR. 2. The quantity of the pressure is the same always at equal dis
tances from the centre, whether the superficies pressed be parallel to the
horizon, or perpendicular, or oblique ; or whether the fluid, continued up
wards from the compressed superficies, rises perpendicularly in a rectilinear
direction, or creeps obliquely through crooked cavities and canals, whether
those passages be regular or irregular, wide or narrow. That the pressure
is not altered by any of these circumstances, may be collected by applying
the demonstration of this Theorem to the several cases of fluids.

COR. 3. From the same demonstration it may also be collected (by Prop.
XIX), that the parts of a heavy fluid acquire no motion among themselvei
by the pressure of the incumbent veight, except that motion which arises
from condensation.

296 THE MATHEMATICAL PRINCIPLES [BCOK II

Con. 4. And therefore if another body of the same specific gravity, in
capable of condensation, be immersed in this fluid, it will acquire no mo
tion by the pressure of the incumbent weight: it will neither descend nor .
ascend, nor change its figure. If it be spherical, it will remain so, notwith
standing the pressure ; if it be square, it will remain square; and that,
whether it be soft or fluid : whether it swims freely in the fluid, or lies at
the bottom. For any internal part of a fluid is in the same state with the
submersed body ; and the case of all submersed bodies that have the same
magnitude, figure, and specific gravity, is alike. If a submersed body, re
taining its weight, should dissolve and put on the form of a fluid, this
body, if before it would have ascended, descended, or from any pressure as
sume a new figure, would now likewise ascend, descend, or put on a new
figure ; and that, because its gravity and the other causes of its motion
remain. But (by Case 5, Prop. XIX; it would now be at rest, and retain
its figure. Therefore also in the former case.

COR. 5. Therefore a body that is specifically heavier than a fluid con
tiguous to it will sink ; and that which is specifically lighter will ascend,
and attain so much motion and change of figure as that excess or defect of
gravity is able to produce. For that excess or defect is the same thing as an
impulse, by which a body, otherwise in equilibria with the parts of the
fluid, is acted on: and may be compared with the excess or defect of a
weight in one of the scales of a balance.

COR. 6. Therefore bodies placed in fluids have a twofold gravity • the
one true and absolute, the other apparent, vulgar, and comparative. Ab
solute gravity is the whole force with which the body tends downwards ;
relative and vulgar gravity is the excess of gravity with which the body
tends downwards more than the ambient fluid. By the first kind of grav
ity the parts of all fluids and bodies gravitate in their proper places ; and
therefore their weights taken together compose the weight of the whole.
For the whole taken together is heavy, as may be experienced in vessels
full of liquor ; and the weight of the whole is equal to the weights of all
the parts, and is therefore composed of them. By the other kind of grav
ity bodies do not gravitate in their places ; that is, compared with one
another, they do not preponderate, but, hindering one another's endeavours
to descend, remain in their proper places, as if they were not heavy. Those
things which are in the air, and do not preponderate, are commonly looked
on as not heavy. Those which do preponderate are commonly reckoned
heavy, in as much as they are not sustained by the weight of the air. The
Common weights are nothing else but the excess of the true weights above
the weight of the air. Hence also, vulgarly, those things are called light
which are less heavy, and, by yielding to the preponderating air, mount
upwards. But these are only comparatively lig s &mA not truly so, because

hey descend in racuo. Thus, in water, bodies *>icfc. by their greater or

SEC. V.] OF NATURAL PHILOSOPHY. 297

less gravity, descend or ascend, are comparatively and apparently heavy or
light ; and their comparative and apparent gravity or levity is the excess
.or defect by which their true gravity either exceeds the gravity of the
water or is exceeded by it. But those things which neither by preponder
ating descend, nor, by yielding to the preponderating fluid, ascend, although
by their true weight they do increase the weight of the whole, yet com
paratively, and in the sense of the vulgar, they do not gravitate in the wa
ter. For these cases are alike demonstrated.

COR. 7. These things which have been demonstrated concerning gravity
take place in any other centripetal forces.

COR. 8. Therefore if the medium in which any body moves be acted on
either by its own gravity, or by any other centripetal force, and the body
be urged more powerfully by the same force ; the difference of the forces is
that very motive force, which, in the foregoing Propositions, I have con
sidered as a centripetal force. But if the body be more lightly urged by
that force, the difference of the forces becomes a centrifugal force, and is tc
be considered as such.

COR. 9. But since fluids by pressing the included bodies do not
change their external figures, it appears also (by Cor. Prop. XIX) that they
will not change the situation of their internal parts in relation to onf
another ; and therefore if animals were immersed therein, and that all sen
sation did arise from the motion of their parts, the fluid will neither hurt
the immersed bodies, nor excite any sensation, unless so far as those bodies
may be condensed by the compression. And the case is the same of any
system of bodies encompassed with a compressing fluid. All the parts of
the system will be agitated with the same motions as if they were placed
in a vacuum, and would only retain their comparative gravity ; unless so
far as the fluid may somewhat resist their motions, or be requisite to con-
glutinate them by compression.

PROPOSITION XXI. THEOREM XVI.

£<et the density of any fluid be proportional to the compression, and its
parts be attracted downwards by a centripetal force reciprocally pro
portional to the distances from the centre : I say, that, if those dis
tances be taken continually proportional, the densities of the fluid at
the same distances will be also continually proportional.
Let ATV denote the spherical bottom of the fluid, S the centre, S A, SB.
SC, SD, SE, SF, &c., distances continually proportional. Erect the per
pendiculars AH, BI, CK, DL, EM, PN, &c., which shall be as the densi
ties of the medium in the places A, B, C, D, E, F : and the specific grav

ATT RT f^K"
ities in those places will be aa -r-, , - &c., or, which is all one, a&-

298

THE MATHEMATICAL PRINCIPLES

[BOOK II.

AH BI CK

ATT BC' CD' Suppose, first, these gravities to be uniformly continued

from A to B, from B to C, from C to D, (fee., the decrements in the points
B, C, D, (fee., being taken by steps. Arid these gravi
ties drawn into the altitudes AB, BC, CD, (fee., will
give the pressures AH, BI, CK, (fee., by which the bot
tom ATV is acted on (by Theor. XV). Therefore the
particle A sustains all the pressures AH, BI, CK, DJL,
(fee., proceeding in infinitum ; and the particle B sus
tains the pressures of all but the first AH ; and the par
ticle C all but the two first AH, BI ; and so on : and
therefore the density AH of the first particle A is to
the density BI of the second particle B as the sum of
all AH -f- BI + CK + DL, in infinitum, to the sum of
all BI -f- CK -f DL, (fee. And BI the density of the second particle B is
to CK the density of the third C, as the sum of all BI -f CK + DL, (fee.,
to the sum of all CK -f- DL, (fee. Therefore these sums are proportional
to their differences AH, BI, CK, (fee., and therefore continually propor
tional (by Lem. 1 of this Book) ; and therefore the differences AH, BI,
CK, (fee., proportional to the sums, are also continually proportional.
Wherefore since the densities in the places A, B, C, (fee., are as AH, BI,
CK, (fee., they will also be continually proportional. Proceed intermis-
sively, and, ex ccquo, at the distances SA, SC, SE, continially proportional,
the densities AH, CK, EM will be continually proportional. And by the
same reasoning, at any distances SA, SD, SG, continually proportional,
the densities AH. DL, GO, will be continually proportional. Let now the
points A, B, C. D, E, (fee., coincide, so that the progression of the specif'.c
gravities from the bottom A to the top of the fluid may be made continual ;
and at any distances SA, SD, SG, continually proportional, the densities
AH, DL, GO, being all along continually proportional, will still remain
continually proportional. Q.E.D.

COR. Hence if the density of the fluid in two places,
as A and E, be given, its density in any other place Q,
may be collected. With the centre S, and the rectan
gular asymptotes SQ, SX, describe an hyperbola cut
ting the perpendiculars AH, EM, QT in «, e, and q}
as also the perpendiculars HX, MY, TZ, let fall upon
the asypmtote SX, in //, 'm, and t. Make the area
Y////Z to the given area YmAX as the given area
EeqQ to the given area EeaA ; and the line Z£ produced will cut off the
line Q,T. proportional to the density. For if the lines SA, SE, SQ are
continually proportional, the areas ReqQ., fyaA will be equal, and thence

X

SEC. V.

OF NATURAL PHILOSOPHY.

299

the areas YwYZ. X/zwY, proportional to them, will be also equal ; and
the lines SX, SY, SZ, that is, AH, EM, Q,T continually proportional, as
they ought to be. And if the lines SA, SE, SQ,5 obtain any other order
in the series of continued proportionals, the lines AH, EM, Q,T, because
of the proportional hyperbolic areas, will obtain the same order in another
series of quantities continually proportional.

PROPOSITION XXII THEOREM XVII.

Let the density of any fluid be proportional to the compression, and its
parts be attracted downwards by a gravitation reciprocally propor
tional to the squares of the distances from the centre : I say, that if
the distances be taken in harmonic progression, the densities of the
fluid at those distances will be in a geometrical progression.
Let S denote the centre, and SA,

SB, SC, SD, SE, the distances in

geometrical progression. Erect the

perpendiculars AH, BI, CK, (fee.,

which shall be as the densities of c

the fluid in the places A, B, C, D, »

E, (fee., and the specific gravities

thereof in those places will be as

AH BI

,^-, (fee. Suppose these

V

LN

M

I.

K

i

V

V

V

V7

^

n

w
r

/

£

L /

/

/t

SA2'SB2'SC2'

gravities to be uniformly continued, the first from A to B, the second from
B to C, the third from C to I), &c. And these drawn into the altitudes
AB, BC, CO, DE, (fec.; or, which is the same thing/into the distances SA,

ATT r>T OT7"

SB, SC, (fee., proportional to those altitudes, will give -~-r-, ^=5, -~~, (fee..

the exponents of the pressures. Therefore since the densities are as th^
sums of those pressures, the differences AH — BI, BI — CK, (fee., of tb,«

densities will be as the differences of those sums ~-r~, ^, ~~, (fee. With

the centre S, and the asymptotes SA, S#, describe any hyperbola, cutting
the perpendiculars AH, BI, CK, (fee., in a, 6, c, (fee., and the perpendicu
lars H/, I//,, K?#, let fall upon the asymptote Sv, in h, i, k ; and the dif
ferences of the densities tu, uw, (fee., will be as „ A , ^^, (fee. And the

SA; SB;

rectangles tu X th, uw X uij (fee., or tp, uq, (fee., as
that is, as Aa, Bb, (fee.

AH X th BI X ui

— , (fee.

SA SB

For, by the nature of the hyperbola, SA is to AH

or St as th to Ar, and therefore — pri — is equal to Aa . And, by a like

SA

300 THE MATHEMATICAL PRINC. PLES [BOOK II.

reasoning, — ^n~~ *s e(lua^ to ^, &c- But Aa> B^> ^c, &cv are continu

ally proportional, and therefore proportional to their differences Aa — B&,
B6 — Cc; &c., therefore the rectangles fy?, nq, &c., are proportional to those
differences ; as also the sums of the rectangles tp + uq, or tp + uq -f w
to the sums of the differences Aa — Cc or Aa — Da7. Suppose several of
these terms, and the sum of all the differences, as Aa — F/, will be pro
portional t? the sum of all the rectangles, as zthn. Increase the number
of terms, and diminish the distances of the points A, B, C, (fee., in iiijini-
tum, and those rectangles will become equal to the hyperbolic area zthn.
and therefore the difference Aa — F/ 19 proportional to this area. Take
nowT any distances, as SA, SD, SF, in harmonic progression, and the dif
ferences Aa — Da7, Da1 — F/ will be equal ; and therefore the areas thlx,
xlnz, proportional to those differences will be equal among themselves, and
the densities St, S:r, Sz, that is, AH, DL, FN, continually proportional.
Q.E.D.

COR. Hence if any two densities of the fluid, as AH and BI, be given,
the area thiu, answering to their difference tu, will be given; and thence
the density FN will be found at any height SF, by taking the area thnz to
that given area thiu as the difference Aa — F/ to the difference Aa — Eh.

SCHOLIUM.

By a like reasoning it may be proved, that if the gravity of the particles
of a fluid be diminished in a triplicate ratio of the distances from the centre ;
and the reciprocals of the squares of the distances SA, SB, SC, &c., (namely,

SA3 SA3 SA3 .

opt' ^e ta^en m an arithmetical progression, the densities AH.

BI, CK, &c., will be in a geometrical progression. And if the gravity be
diminished in a quadruplicate ratio of the distances, and the reciprocals of

the cubes of the distances (as ^-r^, SRS» sps' ^c'^ ^e ta^cn ^n ai>itnmeti-

cai progression, the densities AH, BI, CK, &c., will be in geometrical pro
gression. And so in irtfinitum. Again ; if the gravity of the particles of
the fluid be the same at all distances, and the distances be in arithmetical
progression, the densities will be in a geometrical progression as Dr. Hal-
ley has found. If the gravity be as the distance, and the squares of the
distances be in arithmetical progression, the densities will be in geometri
cal progression. And so in infinitum. These things will be so, when the
density of the fluid condensed by compression is as the force of compres
sion ; or, which is the same thing, when the space possessed by the fluid is
reciprocally as this force. Other laws of condensation may be supposed,
as that the cube of the compressing force may be as the biquadrate of the

SEC. V.] OF NATURAL PHILOSOPHY. 301

de isity ; or the triplicate ratio of tlie force the same with the quadruplicate
ratio of the density : in which case, if the gravity he reciprocally as the
square of the distance from the centre, the density will be reciprocally at
the cube of the distance. Suppose that the cube of the compressing force
be as the quadrato-cube of the density ; and if the gravity be reciprocally
as the square of the distance, the density will be reciprocally in a sesqui-
plicate ratio of the distance. Suppose the compressing force to be in a du
plicate ratio of the density, and the gravity reciprocally in a duplicate ra
tio of the distance, and the density will be reciprocally as 'the distance.
To run over all the cases that might bo offered would be tedious. But as
to our own air, this is certain from experiment, that its density is either
accurately, or very nearly at least, as the compressing force ; and therefore
the density of the air in the atmosphere of the earth is as the weight of
the whole incumbent air, that is, as the height of the mercury in the ba
rometer.

PROPOSITION XXIII. THEOREM XVIII.

If a fluid be composed of particles mutually flying each other, and the
drnsity be as the compression, the centrifugal forces of the particles
'will be reciprocally proportional to tlie distances of their centres. And,
vice versa, particles flying each otli,er, with forces that are reciprocally
proportional to the distances of their centres^ compose an elastic fluid,
whose density is as the compression.
Let the fluid be supposed to be included in a cubic
space ACE, and then to be reduced by compression into
a lesser cubic space ace ; and the distances of the par- F
tides retaining a like situation with respect to each
other in both the spaces, will be as the sides AB, ab of
the cubes ; and the densities of the mediums will be re
ciprocally as the containing spaces AB3, ab3. In the
plane side of the greater cube A BCD take the square
DP equal to the plane side db of the lesser cube: and,
by the supposition, the pressure with which the square
DP urges the inclosed fluid will be to the pressure with
which that square db urges the inclosed fluid as the densities of the me
diums are to each other, that is, asa/>3 to AB3. But the pressure with
which the square DB urges the included fluid is to the pressure with which
the square DP urges the same fluid as the square DB to the square DP,
that is, as AB2 to abz. Therefore, ex cequo, the pressure with which the
square DB urges the fluid is to the pressure with which the square db
urges the fluid as ab to AB. Let the planes FGH,/°V?, U drawn through
the middles of the two cubes, and divide the fluid into tw^/ parts, These
parts will press each other mutually with the same forces with which they

A

THE MATHEMATICAL PRINCIPLES [BOOK II.

are themselves pressed by the planes AC, ac, that is, in the proportion of
ab to AB : arid therefore the centrifugal forces by which these pressures
are sustained are in the same ratio. The number of the particles being
equal, and the situation alike, in both cubes, the forces which all the par
ticles exert, according to the planes FGH,/o7/,, upon all, are as the forces
which each exerts on each. Therefore the forces which each exerts on
each, according to the plane FGH in the greater cube, are to the forces
which each exerts on each, according to the plane fgh in the lesser cube,
us ab to AB,*that is, reciprocally as the distances of the particles from each
other. Q.E.D.

And, vice versa, if the forces of the single particles are reciprocally as
the distances, that is, reciprocally as the sides of the cubes AB, ab ; the
sums of the forces will be in the same ratio, and the pressures of the sides
i)B, db as the sums of the forces ; and the pressure of the square DP to
the pressure of the side DB as ab2 to AB 2. And, ex cequo, the pressure of
the square DP to the pressure of the side db as ab* to AB3 ; that is, the
force of compression in the one to the force of compression in the other as
the density in the former to the density in the latter. Q.E.D.

SCHOLIUM.

By a like reasoning, if the centrifugal forces of the particles are recip
rocally in the duplicate ratio of the distances between the centres, the cubes
of the compressing forces will be as the biquadrates of the densities. If
the centrifugal forces be reciprocally in the triplicate or quadruplicate ratio
of the distances, the cubes of the compressing forces will be as the quadrato-
cubes, or cubo-cubes of the densities. And universally, if D be put for the
distance, and E for the density of the compressed fluid, and the centrifugal
forces be reciprocally as any power Dn of the distance, whose index is the
number ??, the compressing forces will be as the cube roots of the power
En + 2. whose index is the number n + 2 ; and the contrary. All these
things are to be understood of particles whose centrifugal forces terminate
in those particles that are next them, or are diffused not much further.
We have an example of this in magnetical bodies. Their attractive vir
tue is terminated nearly in bodies of their own kind that are next them.
The virtue of the magnet is contracted by the interposition of an iron
plate, and is almost terminated at it : for bodies further off are not attracted
by the magnet so much as by the iron plate. If in this manner particles repel
others of their own kind that lie next them, but do not exert their virtue
on the more remote, particles of this kind will compose such fluids as are
treated of in this Proposition, If the virtue of any particle diffuse itself
every way in inftnitum, there will be required a greater force to produce
an equal condensation of a greater quantity of the flui 1. But whether

SEC. VI.] OF NATURAL PHILOSOPHY. 303

elastic fluids do really consist of particles so repelling each other, is a phy
sical question. We have here demonstrated mathematically the property
of fluids consisting of particles of this kind, that hence philosophers may
take occasion to discuss that question.

SECTION VI.

Of the motion and resistance of funependulous bodies.

PROPOSITION XXIV. THEOREM XIX.

The quantities of matter i/i funependulous bodies, whose centres of oscil
lation are equally distant from, the centre of suspension, are in a, ratio
compounded of the ratio of the weights and the duplicate ratio of the
times of the oscillations in vacuo.

For the velocity which a given force can generate in a given matter in
a given time is as the force and the time directly, and the matter inversely.
The greater the force or the time is, or the less the matter, the greater ve
locity will he generated. This is manifest from the second Law of Mo
tion. Now if pendulums are of the same length, the motive forces in places
equally distant from the perpendicular are as the weights : and therefore
if two bodies by oscillating describe equal arcs, and those arcs are divided
into equal parts ; since the times in which the bodies describe each of the
correspondent parts of the arcs are as the times of the whole oscillations,
the velocities in the correspondent parts of the oscillations will be to each
other as the motive forces and the whole times of the oscillations directly,
and the quantities of matter reciprocally : and therefore the quantities of
matter are as the forces and the times of the oscillations directly and the
velocities reciprocally. But the velocities reciprocally are as the times,
and therefore the times directly and the velocities reciprocally are as the
squares of the times; and therefore the quantities of matter are as the mo
tive forces and the squares of the times, that is, as the weights and the
squares of the times. Q.E.D.

COR. 1. Therefore if the times are equal, the quantities of matter in
each of the bodies are as the weights.

COR. 2. If the weights are equal, the quantities of matter will be as the
pquarcs of the times.

COR. 3. If the quantities of matter are equal, the weights will be recip
rocally as the squares of the times.

COR. 4. Whence since the squares of the times, cceteris paribus, are as
the length* of the pendulums, therefore if both the times and quantities of
matter are equal, the weights will be as the lengths of the pendulums.

J04 THE MATHEMATICAL PRINCIPLES [BOOK 11

COR. 5. And universally, the quantity of matter in the pendulous body
is as the weight and the square of the time directly, and the length of the
pendulum inversely.

COR. 6. But in a non-resisting medium, the quantity of matter in the
pendulous body is as the comparative weight and the square of the time
directly, and the length of the pendulum inversely. For the comparative
weight is the motive force of the body in any heavy medium, as was shewn
above ; and therefore does the same thing in such a non-resisting medium
as the absolute weight does in a vacuum.

COR. 7. And hence appears a method both of comparing bodies one
among another, as to the quantity of matter in each ; and of comparing
the weights of the same body in different places, to know the variation of
its gravity. And by experiments made with the greatest accuracy, I
have always found the quantity of matter in bodies to be proportional to
their weight.

PROPOSITION XXV. THEOREM XX.

Funependulous bodies that are, in, any 'medium, resisted in the ratio oj
the moments of time, and funepetidulons bodies that move in a non-
resisting medium of the same specific gravity, perform their oscilla
tions in. a cycloid in the same time, and describe proportional parts oj
arcs together.

Let AB be an arc of a cycloid, which
a body D, by vibrating in a non-re
sisting medium, shall describe in any
time. Bisect that arc in C, so that C
may be the lowest point thereof ; and
the accelerative force with which the
body is urged in any place D, or d or
E, will be as the length of the arc CD,

pressed by that same arc ; and since the resistance is as the moment of the
time, and therefore given, let it ba expressed by the given part CO of the
cycloidal arc, and take the arc Od in the same ratio to the arc CD that
the arc OB has to the arc CB : and the force with which the body in d is
urged in a resisting medium, being the excess of the force Cd above the
resistance CO, will be expressed by the arc Od, and will therefore be to
the force with which the body D is urged in a non-resisting medium in the
place D, as the arc Od to the arc CD ; and therefore also in the place B,
as the arc OB to the arc CB. Therefore if two bodies D, d go from the place
B, and are urged by these forces ; since the forces at the beginning are as
the arc CB and OB, the first velocities and arcs first described will be in
the same ratio. Let those arcs be BD and Ed, and the remaining arcf

SEC. VI. | OF NATURAL PHILOSOPHY. 305

CD, Odj will be in the same ratio. Therefore the forces, being propor
tional to those arcs CD, Od, will remain in the same ratio as at the be
ginning, and therefore the bodies will continue describing together arcs in
the same ratio. Therefore the forces and velocities and the remaining arcs
CD. Od, will be always as the whole arcs CB, OB, and therefore those re
maining arcs wLl be described together. Therefore the two bodies D and
d will arrive together at the places C and O ; that whicli moves in the
non-resisting medium, at the place C, and the other, in the resisting me
dium, at the place O. Now since the velocities in C and O are as the arcs
CB, OB, the arcs which the bodies describe when they go farther will be
in the same ratio. Let those arcs be CE and Oe. The force with which
the body D in a non-resisting medium is retarded in E is as CE, and the
force with which the body d in the resisting medium is retarded in e, is as
the sum of the force Ce and the resistance CO, that is, as Oe ; and there
fore the forces with which the bodies are retarded are as the arcs CB, OB,
proportional to the arcs CE, Oe ; and therefore the velocities, retarded in
that given ratio, remain in the same given ratio. Therefore the velocities
and the arcs described with those velocities are always to each other in
that given ratio of the arcs CB and OB ; and therefore if the entire arcs
AB, aB are taken in the same ratio, the bodies D andc/ will describe those
aics together, and in the places A and a will lose all their motion together.
Therefore the whole oscillations are isochronal, or are performed in equal
times ; and any parts of the arcs, as BD, Ed, or BE, Be, that are described
together, are proportional to the whole arcs BA, B«. Q,.E.D.

COR. Therefore the swiftest motion in a resisting medium does not fall
upon the lowest point C, but is found in that point O, in which the whole
arc described Ba is bisected. And the body, proceeding from thence to a,
is retarded at the same rate with which it was accelerated before in its de
scent from B to O.

PROPOSITION XXVI. THEOREM XXI.

Funependulous bodies, that are resisted in the ratio of the velocity, have

their oscillations in a cycloid isochronal.

For if two bodies, equally distant from their centres of suspension, de
scribe, in oscillating, unequal arcs, and the velocities in the correspondent
parts of the arcs be to each other as the whole arcs ; the resistances, pro
portional to the velocities, will be also to each other as the same arcs.
Therefore if these resistances be subducted from or added to the motive
forces arising from gravity which are as the same arcs, the differences or
sums will be to each other in the same ratio of the arcs ; and since the in
crements and decrements of the velocities are as these differences or sums,
the velocities will be always as the whole arcs; therefore if the velocities
are in any one case as the whole arcs, they will remain always in the same

20

306 THE MATHEMATICAL PRINCIPLES [BOOK. 11

ratio. But at the beginning of the motion, when the bodies begin to de
scend and describe those arcs, the forces, which at that time are proportional
to the arcs, will generate velocities proportional to the arcs. Therefore
the velocities will be always as the whole arcs to be described, and there
fore those arcs will be described in the same time. Q,.E.D.

PROPOSITION XXVII. THEOREM XXII.

If fnnependulous bodies are resisted in the duplicate ratio of their
velocities, the differences between the times of the oscillations in a re
sisting medium, and the times of the oscillations in a non-resisting
medium of the same specific gravity, will be proportional to the arcs
described in oscillating nearly.

For let equal pendulums in a re
sisting medium describe the unequal
arcs A, B ; and the resistance of the
body in the arc A will be to the resist
ance of the body in the correspondent
part of the arc B in the duplicate ra
tio of the velocities, that is, as, A A to
BB nearly. If the resistance in the
arc B were to the resistance in the arc
A as AB to AA, the times in the arcs A and B would be equal (by the last
Prop.) Therefore the resistance AA in the arc A, or AB in the arc B,
causes the excess of the time in the arc A above the time in a non-resisting
medium ; and the resistance BB causes the excess of the time in the arc B
above the time in a non-resisting medium. But those excesses are as the
efficient forces AB and BB nearly, that is, as the arcs A and B. Q.E.D.

COR, 1. Hence from the times of the oscillations in unequal arcs in a
resisting medium, may be knowrn the times of the oscillations in a non- re
sisting medium of the same specific gravity. For the difference of the
times will be to the excess of the time in the lesser arc above the time in a
non-resisting medium as the difference of the arcs to the lesser arc.

COR. 2. The shorter oscillations are more isochronal, and very short
ones are performed nearly in the same times as in a non-resisting medium.
But the times of those which are performed in greater arcs are a little
greater, because the resistance in the descent of the body, by which the
time is prolonged, is greater, in proportion to the length described in the
descent than the resistance in the subsequent ascent, by which the time is
contracted. But the time of the oscillations, both short arid long, seems to
be prolonged in some measure by the motion of the medium. For retard
ed bodies are resisted somewhat less in proportion to the velocity, and ac
celerated bodies somewhat more than those that proceed uniformly forwards ;

SEC. VI.]

OF NATURAL PHILOSOPHY.

307

because the medium, by the motion it has received from the bodies, going
forwards the same way with them, is more agitated in the former case, and
less in the latter ; and so conspires more or less with the bodies moved.
Therefore it resists the pendulums in their descent more, and in their as
cent less, than in proportion to the velocity; and these two causes concur
ring prolong the time.

PROPOSITION XXVIII. THEOREM XXIII.

If afunependvlous body, oscillating in a cycloid, be resisted in the rati >
of the moments of the time, its resistance will be to the force of grav
ity as the excess of the arc described in the whole descent above the
arc described in the subsequent ascent to twice the length of the pen
dulum.

Let BC represent the arc described
in the descent, Ca the arc described in
the ascent, and Aa the difference of
the arcs : and things remaining as they
were constructed and demonstrated in
Prop. XXV, the force with which the
oscillating body is urged in any place
D will be to the force of resistance as
the arc CD to the arc CO, which is
half of that difference Aa. Therefore the force with which the oscillating
body is urged at the beginning or the highest point of the cycloid, that is,
the force of gravity, will be to the resistance as the arc of the cycloid, be
tween that highest point and lowest point C, is to the arc CO ; that is
(doubling those arcs), as the whole cycloidal arc, or twice the length of the
pendulum, to the arc Aa. Q.E.D.

PROPOSITION XXIX. PROBLEM VI.

Supposing that a body oscillating in a cycloid is resisted in a duplicate

ratio of the velocity: to find the resistance in each place.
Let Ba be an arc described in one entire oscillation, C the lowest point

C O

K

O ,S P rR Q M

of the cycloid, and CZ half the whole cycloidal arc, equal to the length of
the pendulum ; and let it be required to find the resistance of the body is

30S THE MATHEMATICAL PRINCIPLES [BOOK 1L

any place D. Cut the indefinite right line OQ in the points O, S, P, Q,,
so that (erecting the perpendiculars OK, ST, PI, QE, and with the centre
O, and the aysmptotcs OK, OQ, describing the hyperbola TIGE cutting
the perpendiculars ST, PI, QE in T. I, and E, and through the point I
drawing KF. parallel to the asymptote OQ, meeting the asymptote OK i i
K, and the perpendiculars ST and QE in L and F) the hyperbolic area
PIEQ may be to the hyperbolic area PITS as the arc BC, described in the
descent of the body, to the arc Ca described in the ascent ; and that the
area IEF may be to the area ILT as OQ to OS. Then with the perpen
dicular MN cut off the hyperbolic area PINM, and let that area be to the
hyperbolic area PIEQ as the arc CZ to the arc BC described in the de
scent. And if the perpendicular RG cut off the hyperbolic area PIGR,
which shall be to the area PIEQ as any arc CD to the arc BC described
in the whole descent, the resistance in any place D will be to the force of
OR

gravity as the area IEF — IGH to the area PINM.

For since the forces arising from gravity with which the body is
urged in the places Z, B, D, a, are as the arcs CZ. CB, CD, Ca and those
arcs are as the areas PINM, PIEQ, PIGR, PITS; let those areas be the
exponents both of the arcs and of the forces respectively. Let DC? be a
very small space described by the body in its descent : and let it be expressed
r>y the very small area RGor comprehended between the parallels RG, rg ;
and produce r<? to //, so that GYlhg- and RG«r may be the contemporane
ous decrements of the areas IGH, PIGR. And the increment Gllhg —

IEF, or Rr X HG — -^ IEF, of the area ~ IEF —IGH will be

,
OQ OQ

IFF

to the decrement RG°r, or Rr X RG, of the area PIGR, as HG — - -

OR

to RG ; and therefore as OR X HG — IEF to OR X GR or OP X

PL that is (because of the equal quantities OR X HG, OR X HR — OR
X GR, ORHK — OPIK, PIHR and PIGR + IGH), as PIGR + IGH —
OR OR

IEF to OPIK. Therefore if the area - IEF — IGH be called

OQ

Y, and RGgr the decrement of the area PIGR be given, the increment of

the area Y will be as PIGR — Y.

Then if V represent the force arising from the gravity, proportional to
the arc CD to be described, by which the body is acted upon in D, and R
be put for the resistance, V — R will be the whole force with which the
body is urged in D. Therefore the increment of the velocity is as V — R
and the particle of time in which it is generated conjunctly. But the ve
locity itself is as the contempo] aueous increment of the space described di-

SEC. VI.J

OF NATURAL PHILOSOPHY.

309

rectly and the same particle of time inversely. Therefore, since the re
sistance is, by the supposition, as the square of the velocity, the increment
of the resistance will (by Lem. II) be as the velocity and the increment of
the velocity conjunctly, that is, as the moment of the space and V — R
conjunctly ; and, therefore, if the moment of the space be given, as V —
11 ; that is, if for the force V we put its exponent PIGR, and the resist
ance R be expressed by any other area Z; as PIGR — Z.v

Therefore the area PIGR uniformly decreasing by the subduction of
given moments, the area Y increases in proportion of PIGR — Y, and
the area Z in proportion of PIGR — Z. And therefore if the areas
Y and Z begin together, and at the beginning are equal, these, by the
addition of equal moments, will continue to be equal • and in like man
ner decreasing by equal moments, \vill vanish together. And, vice versa,
if they together begin and vanish, they will have equal moments and te
always equal ; and that, because if the resistance Z be augmented, the ve
locity together with the arc C«, described in the ascent of the body, will be
diminished ; and the point in which all the motion together with the re
sistance ceases coming nearer to the point C, the resistance vanishes sooner
than the area Y. And the contrary will happen when the resistance is
diminished.

Now the area Z begins and ends where the resistance is nothing, that is,
at the beginning of the motion where the arc CD is equal to the arc CB,

K

/IK

O S P /~R Q M

and the right line RG falls upon the right line Q.E ; and at the end of
the motion where the arc CD is equal to the arc Ca, and RG falls upon

the right line ST. And the area* Y or — IEF — IGH begins and ends

also where the resistance is nothing, and therefore where IEF and

IGH are equal ; that is (by the construction), where the right line RG
falls successively upon the right lines Q,E and ST. Therefore those areas
begin and vanish together, and are therefore always equal. Therefore the area
OR

IEF — IGH is equal to the area Z, by which the resistance is ex

pressed, and therefore is to the area PINM, by which the gravity is ex
pressed, as the resistance to the gravity. Q.E.D.

310

THE MATHEMATICAL PRINCIPLES

[BOOK 11.

COR. 1 . Therefore the resistance in the lowest place C is to the force

OP

of gravity as the area ^ ~ IEF to the area PINM.

COR. 2. But it becomes greatest where the area PIHR is to the area
IEF as OR to OQ. For in that case its moment (that is, PIGR — Y)
becomes nothing.

COR. 3. Hence also may be known the velocity in each place, as being
in the subduplicate ratio of the resistance, and at the beginning of the mo
tion equal to the velocity of the body oscillating in the same cycloid with
out any resistance.

However, by reason of the difficulty of the calculation by which the re
sistance and the velocity are found by this Proposition, we have thought
fit to subjoin the Proposition following.

PROPOSITION XXX. THEOREM XXIV.

If a right line aB be equal to the arc of a cycloid which an oscillating
body describes, and at each of its points D the perpendiculars DK be
erected, which shall be to the length of the pendulum as the resistance
of the body in the corresponding points of the arc to the force of grav
ity ; I say, that the difference between the arc described in the whole
descent and the arc described in the whole subsequent ascent drawn
into half the sum of the same arcs will be equal to the area BKa
which all those perpendiculars take up.

Let the arc of the cycloid, de
scribed in one entire oscillation, be
expressed by the right line aB,
equal to it, and the arc which
would have been described in vaciw
by the length AB. Bisect AB in
C, and the point C will represent
the lowest point of the cycloid, and
CD Mill be as the force arising from gravity, with which the body in D i,s
urged in the direction of the tangent of the cycloid, and will have the same
ratio to the length of the pendulum as the force in D has to the force of
gravity. Let that force, therefore, be expressed by that length CD, and
the force of gravity by the length of the pendulum ; and if in DE you
take DK in the same ratio to the length of the pendulum as the resistance
has to the gravity, DK will be the exponent of the resistance. From the
centre C with the interval CA or CB describe a semi-circle BEeA. Let
the body describe, in the least time, the space Dd ; and, erecting the per
pendiculars DE, de, meeting the circumference in E and e, they will be as
the velocities which the body descending in vacuo from the point B would
acquire in the places D and d. This appears by Prop, LII, Book L Let

SEC. VLJ OF NATURAL PHILOSOPHY. 311

therefore, these velocities be expressed by those perpendiculars DE, de ;
arid let DF be the velocity which it acquires in D by falling from B in
the resisting medium. And if from the centre C with the interval OF we
describe the circle F/M meeting the right lines de and AB in / and M,
then M will be the place to which it would thenceforward, without farther
resistance, ascend, and (//"the velocity it would acquire in d. Whence,
also, if FO- represent the moment of the velocity which the body D, in de
scribing the least space DC/, loses by the resistance of the medium ; and
CN be taken equal to Cg ; then will N be the place to which the body, if
it met no farther resistance, would thenceforward ascend, and MN will be
the decrement of the ascent arising from the loss of that velocity. Draw
F/n perpendicular to dft and the decrement Fg of the velocity DF gener
ated by the resistance DK will be to the increment fm of the same velo
city, generated by the force CD, as the generating force DK to the gener
ating force CD. But because of the similar triangles F////, Fhg, FDC,
fm is to Fm or Dd as CD to DF ; and, ex ceqno, Fg to Dd as DK to
DF. Also Fh is to Fg as DF to CF ; and, ex ax/uo perturbate, Fh or
MN to Do1 as DK to CF or CM ; and therefore the sum of all the MN X
CM will be equal to the sum of all the Dd X DK. At the moveable
point M suppose always a rectangular ordinate erected equal to the inde
terminate CM, which by a continual motion is drawn into the whole
length Aa ; and the trapezium described by that motion, or its equal, the
rectangle Aa X |aB, will be equal to the sum of all the MN X CM, and
therefore to the sum of all the Dd X DK, that is, to the area BKVTa

O.E.D.

COR. Hence from the law of resistance, and the difference Aa of the
arcs Ca} CB, may be collected the proportion of the resistance to the grav
ity nearly.

For if the resistance DK be uniform, the figure BKTa will be a rec
tangle under Ba and DK; and thence the rectangle under ^Ba and Aa
will be equal to the rectangle under Ba and DK, and DK will be equal to
jAa. Wherefore since DK is the exponent of the resistance, and the
length of the pendulum the exponent of the gravity, the resistance will be
to the gravity as \Aa to the length of the pendulum ; altogether as in
Prop. XXVIII is demonstrated.

If the resistance be as the velocity, the figure BKTa will be nearly an
ellipsis. For if a body, in a non-resisting medium, by one entire oscilla
tion, should describe the length BA, the velocity in any place D would be
as the ordinate DE of the circle described on the diameter AB. There
fore since Ea in the resisting medium, and BA in the non-resisting one,
are described nearly in the same times ; and therefore the velocities in each
of the points of Ba are to the velocities in the correspondent points of the
length BA. nearly as Ba is to BA , the velocity in the point D in the re-

312

THE MATHEMATICAL PRINCIPLES

[BJOK 11.

sisting medium will be as the ordinate of the circle or ellipsis described
upon the diameter Ba ; and therefore the figure BKVTa will be nearly ac
ellipsis. Since the resistance is supposed proportional to the velocity, le\
OV be the exponent of the resistance in the middle point O ; and an ellip
sis BRVSa described with the centre O, and the semi-axes OB, OV, will
be nearly equal to the figure BKVTa, and to its equal the rectangle Act
X BO. Therefore Aa X BO is to OV X BO as the area of this ellipsis
to OV X BO; that is, Aa is to OV as the area of the semi-circle to the
square of the radius, or as 1 1 to 7 nearly ; and, therefore, T7T Aa is to the
length of the pendulum as the resistance of the oscillating body in O to
its gravity.

Now if the resistance DK be in the duplicate ratio of the velocity, the
figure BKVTa will be almost a parabola having V for its vertex arid OV
for its axis, and therefore will be nearly equal to the rectangle under f Ba
and OV. Therefore the rectangle under |Ba and Aa is equal to the rec
tangle f Ba X OV, and therefore OV is equal to f Aa ; and therefore the
resistance in O made to the oscillating body is to its gravity as f Aa to the
length of the pendulum.

And I take these conclusions to be accurate enough for practical uses.
For since an ellipsis or parabola BRVSa falls in with the figure BKVTa
in the middle point V, that figure, if greater towards the part BRV or
VSa than the other, is less towards the contrary part, and is therefore
nearly equal to it.

PROPOSITION XXXI. THEOREM XXV.

If the resistance made to an oscillating body in each of the proportional
parts of the arcs described be augmented or diminished in, a given ra
tio, the difference between the arc described in the descent and the arc
described in the subsequent ascent ivill be augmented or diminished in
the same ratio.

For that difference arises from
the retardation of the pendulum
by the resistance of the medium,
and therefore is as the whole re
tardation and the retarding resist
ance proportional thereto. In the
foregoing Proposition the rectan-

M isr u c o «.-/ n P gle under the right line ^aB and

the difference Aa of the arcs CB, Ca, was equal to the area BKTa, And
that area, if the length aB remains, is augmented or diminished in the ra
tio of the ordinates DK ; that is, in the ratio of the resistance and is there
fore as the length aB and the resistance conjunctly. And therefore the
rectangle under Aa and |aB is as aB and the resistance conjunctly, anc
therefore Aa is as the resistance. QJE.D.

SEC. VI.l OF NATURAL PHILOSOPHY. 313

COR. 1. Hence if the resistance be as the velocity, the difference of
the arts in the same medium will be as the whole arc described : and the
contrary.

COR. 2. If the resistance be in the duplicate ratio of the velocity, that
difference will be in the duplicate ratio of the whole arc : and the contrary.

COR. 3. And universally, if the resistance be in the triplicate or any
other ratio of the velocity, the difference will be in the same ratio of the.
whole arc : and the contrary.

COR. 4. If the resistance be partly in the simple ratio of the velocity,
and partly in the duplicate ratio of the same, the difference will be partly
in the ratio of the whole arc, and partly in the duplicate ratio of it: and
the contrary. So that the law arid ratio of the resistance will be the
same for the velocity as the law and ratio of that difference for the length
of the arc.

COR. 5. And therefore if a pendulum describe successively unequal arcs,
and we can find the ratio of the increment or decrement of this difference
for the length of the arc described, there will be had also the ratio of the
increment or decrement of the resistance for a greater or less velocity.

GENERAL SCHOLIUM.

From these propositions we may find the resistance of mediums by pen
dulums oscillating therein. I found the resistance of the air by the fol
lowing experiments. I suspended a wooden globe or ball weighing oT^
ounces troy, its diameter CJ London inches, by a fine thread on a firm
hook, so that the distance between the hook and the centre of oscillation of
the globe was 10| feet. I marked on the thread a point 10 feet and 1 inch
distant from the centre of suspension • and even with that point I placed a
ruler divided into inches, by the help whereof I observed the lengths of the
arcs described by the pendulum. Then I numbered the oscillations ia
which the globe would lose -{- part of its motion. If the pendulum was
drawn aside from the perpendicular to the distance of 2 inches, and thence
let go, so that in its whole descent it described an arc of 2 inches, and in
the first whole oscillation, compounded of the descent and subsequent
ascent, an arc of almost 4 inches, the same in 164 oscillations lost j part
of its motion, so as in its last ascent to describe an arc of If inches. If
in the first descent it described an arc of 4 inches, it lost j part of its mo
tion in 121 oscillations, so as in its last ascent to describe an arc of 3^
inches. If in the first descent it described an arc of 8, 16, 32, or 64 inches,
it lost | part of its motion in 69, 35|, 18|-7 9| oscillations, respectively.
Therefore the difference between the arcs described in the first descent and
the last ascent was in the 1st, 2d, 3d, 4th, 5th, 6th cases, }, 1. 1, 2, 4, 8
inches respectively. Divide those differences by the number of oscillations
in each case, and in one mean oscillation, wherein an arc of 3£ , 7-|, 15, 30

314 THE MATHEMATICAL PRINCIPLES [BOOK Jl.

60, 120 inches was described, the difference of the arcs described in the
descent and subsequent ascent will be ¥|^, ^{^ e\> T4r; -sji fir parts of an
inch, respectively. But these differences in the greater oscillations are in
the duplicate ratio of the arcs described nearly, but in lesser oscillations
something greater than in that ratio ; and therefore (by Cor. 2, Prop. XXXI
of this Book) the resistance of the globe, when it moves very swift, is in
the duplicate ratio of the velocity, nearly; and when it moves slowly,
somewhat greater than in that ratio.

Now let V represent the greatest velocity in any oscillation, and let A,
B, and C be given quantities, and let us suppose the difference of the arcs

3^

to be AV + BV2 + CV2. Since the greatest velocities are in the cycloid
as ^ the arcs described in oscillating, and in the circle as | the chords of
those arcs ; and therefore in equal arcs are greater in the cycloid than in
the circle in the ratio of | the arcs to their chords ; but the times in the
circle are greater than in the cycloid, in a reciprocal ratio of the velocity ;
it is plain that the differences of the arcs (which are as the resistance and
the square of the time conjunctly) are nearly the same in both curves : for
in the cycloid those differences must be on the one hand augmented, with
the resistance, in about the duplicate ratio of the arc to the chord, because
of the velocity augmented in the simple ratio of the same ; and on the
other hand diminished, with the square of the time, in the same duplicate
ratio. Therefore to reduce these observations to the cycloid, we must take
the same differences of the arcs as were observed in the circle, and suppose
the greatest velocities analogous to the half, or the whole arcs, that is, to
the numbers ±, 1, 2, 4, 8, 16. Therefore in the 2d, 4th, and 6th cases, put
1,4, and 1 6 for V ; and the difference of the arcs in the 2d case will become

i 2

* = A + B + C; in the4th case, ^- = 4A + SB + 160 ; in the 6th

121 OOj

case, ^ = 16A + 64B -f- 256C. These equations reduced give A =

9?
0,000091 6, B =-. 0,0010847, and C = 0,0029558. Therefore the difference

of the arcs is as 0,0000916V -f 0,0010847V* + 0,0029558 V* : and there
fore since (by Cor. Prop. XXX, applied to this case) the re.-ist;mcc of the
globe in the middle of the arc described in oscillating, where the velocity

is V, is to its weight as T7TAV -f- T\BV^ + f CV2 to the length of the
pendulum, if for A, B, and C you put the numbers found, the resistance of

the globe will be to its weight as 0,0000583V + 0,0007593V* + 0,OJ22169V2
to the length of the pendulum between the centre of suspension and the
ruler, that is, to 121 inches. Therefore since V in the second case repre
sents 1, in the 4th case 4, and in the 6th case 16, the resistance will be to
the weight of the globe, in the 2d case, as 0,0030345 to 121 ; in the 4th, as
0,041748 to 121 ; in the 6th, as 0,61705 to 121.

SEC. VI.] OF NATURAL PHILOSOPHY. 315

The arc, which the point marked in the thread described in the 6th case,

was of 120 — Q^, or 119/g inches. And therefore since the radius was

ya

121 inches, and the length of the pendulum between the point of suspen
sion and the centre of the globe was 126 inches, the arc which the centre of
the globe described was 124/T inches. Because the greatest velocity of the
oscillating body, by reason of the resistance of the air, does not fall on the
lowest point of the arc described, but near the middle place of the whole
arc, this velocity will be nearly the same as if the globe in its whole descent
in a non-resisting medium should describe 62 ^ inches, the half of that arc,
and that in a cycloid, to which we have above reduced the motion of the
pendulum; and therefore that velocity will be equal to that which the
globe would acquire by falling perpendicularly from a height equal to the
versed sine of that arc. But that versed sine in the cycloid is to that arc
62/2 as the same arc to twice the length of the pendulum 252, and there
fore equal to 15,278 inches. Therefore the velocity of the pendulum is the
same which a body would acquire by falling, and in its fall describing a
space of 15,278 inches. Therefore with such a velocity the globe meets
with a resistance which is to its weight as 0,61705 to 121, or (if we take
that part only of the resistance which is in the duplicate ratio of the ve-
loc.ty) as 0,56752 to 121.

I found, by an hydrostatical experiment, that the weight of this wooden
globe was to the weight of a globe of water of the same magnitude as 55
to 97: and therefore since 121 is to 213,4 in the same ratio, the resistance
made to this globe of water, moving forwards with the above-mentioned
velocity, will be to its weight as 0,56752 to 213,4, that is, as 1 to 376^.
Whence since the weight of a globe of water, in the time in which the
globe with a velocity uniformly continued describes a length of 30,556
inches, will generate all that velocity in the falling globe, it is manifest
that the force of resistance uniformly continued in the same time will take
away a velocity, which will be less than the other in the ratio of 1 to 376^-0,

that is, the rr^-r part of the whole velocity. And therefore in the time

37 VSG

Jiat the globe, with the same velocity uniformly continued, would describe
the length of its semi-diameter, or 3T\ inches, it would lose the 3^42 part
of its motion.

I also counted the oscillations in which the pendulum lost j part of its
motion. In the following table the upper numbers denote the length of the
arc described in the first descent, expressed in inches and parts of an inch ;
the middle numbers denote the length of the arc described in the last as
cent ; and in the lowest place are the numbers of the oscillations. I give
un account of this experiment, as being more accurate than that in which

316 THE MATHEMATICAL PRINCIPLES [BOOK ll

only 1 part of the motion was lost. I leave the calculation to such as are
disposed to make it.

First descent ... 2 4 8 16 32 64

Last ascent . . , 1| 3 6 12 24 48

Numb.ofoscilL . .374 272 162i 83J 41f 22|

I afterward suspended a leaden globe of 2 inches in diameter, weighing
26 1 ounces troy by the same thread, so that between the centre of the
globe and the point of suspension there was an interval of 10^ feet, and 1
counted the oscillations in which a given part of the motion was lost. The
iirst of the following tables exhibits the number of oscillations in which J-
part of the whole motion was lost ; the second the number of oscillations
in which there was lost \ part of the same.

First descent .... 1 2 4 8 16 32 64

Last ascent .... f ' J 3^ 7 14 28 56

Numb, of oscilL . . 226 228 193 140 90^ 53 30

First descent .... 1 2 4 8 16 32 64

Last ascent .... £ 1^ 3 6 12 24 4S

Nunib. of oscill. . .510 518^ 420 318 204 12170

Selecting in the first table the 3d, 5th, and 7th observations, and express
ing the greatest velocities in these observations particularly by the num
bers 1, 4, 16 respectively, 'and generally by the quantity V as above, there

will come out in ihe 3d observation ~- = A + B + C, in the 5th obser-

2 8

vation ^— = 4A 4- 8B + 16C. in the 7th observation ^-- == 16A 4- 64B t-
,t(j j oU

256C. These equations reduced give A = 0,001414, B == 0,000297, C —
0,000879. And thence the resistance of the globe moving with the velocity
V will be to its weight 26^ ounces in the same ratio as 0,0009V +

0,000208V* + 0,000659V2 to 121 inches, the length of the pendulum.
And if we regard that part only of the resistance which is in the dupli
cate ratio of the velocity, it will be to the weight of the globe as 0,000659V2
to 121 inches. But this part of the resistance in the first experiment was
to the weight oi' the wooden globe of 572-72 ounces as 0,002217V2 to 121 ;
and thence the resistance of the wooden globe is to the resistance of the
leaden one (their velocities being equal) as 57/2- into 0,002217 to 26 J-
into 0,000659, that is, as 7|- to 1. The diameters of the two globes were
6f and 2 inches, and the squares of these are to each other as 47 £ and 4,
or 11-J-f and 1, nearly. Therefore the resistances of these equally swift
globes were in less than a duplicate ratio of the diameters. But we have
not yet considered the resistance of the thread, which was certainly very
considerable, and ought to be subducted from the resistance of the pendu
lums here found. I could not determine this accurately, but I found il

SEC. VI.J OF NATURAL PHILOSOPHY. 3 1/

greater than a third part of the whole resistance of the lesser pendulum ;
and thence I gathered that the resistances of the globes, when the resist
ance of the thread is subducted, are nearly in the duplicate ratio of their
diameters. For the ratio of 7} — } to 1 — £, or l(H to 1 is not very
different from the duplicate ratio of the diameters 1 L}f to I.

Since the resistance of the thread is of less moment in greater globes, I
tried the experiment also with a globe whose diameter was ISf inches.
The length of the pendulum between the point of suspension and the cen
tre of oscillation was 122| inches, and between the point of suspension and
the knot in the thread 109| inches. The arc described by the knot at the
first descent of the pendulum was 32 inches. The arc described by the
same knot in the last ascent after five oscillations was 2S inches. The
sum of the arcs, or the whole arc described in one mean oscillation, was 60
inches. The difference of the arcs 4 inches. The y1,,- part of this, or the
difference between the descent and ascent in one mean oscillation, is f of
an inch. Then as the radius 10(J| to the radius 122^, so is the whole arc
of 60 inches described by the knot in one mean oscillation to the whole arc
of 67} inches described by the centre of the globe in one mean oscillation ;
and so is the difference | to a new difference 0,4475. If the length of the
arc described were to remain, and the length of the pendulum should be
augmented in the ratio of 126 to 122}, the time of the oscillation would
be augmented, and the velocity of the pendulum would be diminished in
the subduplicate of that ratio ; so that the difference 0,4475 of the arcs de
scribed in the descent and subsequent ascent would remain. Then if the
arc described be augmented in the ratio of 124 33T to 67}, that difference
0.4475 would be augmented in the duplicate of that ratio, and so would
become 1,5295. These things would be so upon the supposition that the
resistance of the pendulum were in the duplicate ratio of the velocity.
Therefore if the pendulum describe the whole arc of 12433T inches, and its
length between the point of suspension and the centre of oscillation be 126
inches, the difference of the arcs described in the descent and subsequent
ascent would be 1,5295 inches. And this difference multiplied into the
weight of the pendulous globe, which was 208 ounces, produces 318,136.
Again ; in the pendulum above-mentioned, made of a wooden globe, when
its centre of oscillation, being 126 inches from the point of suspension, de
scribed the whole arc of 124 /T inches, the difference of the arcs described

in the descent and ascent was ^^ into ^. This multiplied into the

i/wi y^

weight of the globe, which was 57-272 ounces, produces 49,396. But I mul
tiply these differences into the weights of the globes, in order to find their
resistances. For the differences arise from the resistances, and are as the
resistances directly and the weights inversely. Therefore the resistances
are as the numbers 318,136 and 49,396. But that part of the resistance

31 S THE MATHEMATICAL PRINCIPLES [BOOK 1L

of the lesser globe, which is in the duplicate ratio of the velocity, was to
the whole resistance as 0,56752 to- 0,61675, that is, as 45,453 to 49,396 ;
whereas that part of the resistance of the greater globe is almost equal to
its whole resistance ; and so those parts are nearly as 318,136 and 45,453,
that is, as 7 and 1. But the diameters of the globes are 18f and 6| ; and
their squares 351T9¥ and 47 £J are as 7,438 and 1, that is, as the resistances
of the globes 7 and 1, nearly. The difference of these ratios is scarce
greater than may arise from the resistance of the thread. Therefore those
parts of the resistances which are, when the globes are equal, as the squares
of the velocities, are also, when the velocities are equal, as the squares of
the diameters of the globes.

But the greatest of the globes I used in these experiments was not per
fectly spherical, and therefore in this calculation I have, for brevity's sake,
neglected some little niceties ; being not very solicitous for an accurate
calculus in an experiment that was not very accurate. So that I could
wish that these experiments were tried again with other globes, of a larger
size, more in number, and more accurately formed ; since the demonstra
tion of a vacuum depends thereon. If the globes be taken in a geometrical
proportion, as suppose whose diameters are 4, 8, 16, 32 inches; one may
collect from the progression observed in the experiments what would hap
pen if the globes were still larger.

In order to compare the resistances of different fluids with each other, 1
made the following trials. I procured a wooden vessel 4 feet long, 1 foot
broad, and 1 foot high. This vessel, being uncovered, 1 filled with spring
water, and, having immersed pendulums therein, I made them oscillate in
the water. And I found that a leaden globe weighing 166| ounces, and in
diameter 3f inches, moved therein as it is set down in the following table ;
the length of the pendulum from the point of suspension to a certain
point marked in the thread being 126 inches, and to the centre of oscilla
tion 134 f inches.
The arc described in }

the first descent, by

a point marked in \ 64 . 32 . 16 . $ . 4 . 2 . 1 . £ . J

the thread was \

inches.
The arc described in )

the last ascent was V 48 . 24 . 12 . 6 . 3 . 1| . £ . f . T\

inches. \

The difference of the

arcs, proportional

to the 'motion lost,

was inches.
The number of the os
cillations in water.
The number of the os
cillations in air.

16

«. li . 3 . 7 . lH.12f.13j

85i . 287 . 535

SEC. VI.] OF NATURAL PHILOSOPHY. 319

In the experiments of the 4th column there were equal motions lost in
535 oscillations made in the air, and If in water. The oscillations in the
air were indeed a little swifter than those in the water. But if the oscil
lations in the water were accelerated in such a ratio that the motions of
the pendulums might be equally swift in both mediums, there would be
still the same number 1 j of oscillations in the water, and by these the
same quantity of motion would be lost as before ; because the resistance i>
increased, and the square of the time diminished in the same duplicate ra
tio. The pendulums, therefore, being of equal velocities, there were equal
motions lost in 535 oscillations in the air, and 1} in the water; and there
fore the resistance of the pendulum in the water is to its resistance in the
air as 535 to 1 }. This is the proportion of the whole resistances in the
case of the 4th column.

Now let AV + CV2 represent the difference of the arcs described in the
descent and subsequent ascent by the globe moving in air with the greatest
velocity V ; and since the greatest velocity is in the case of the 4th column
to the greatest velocity in the case of the 1st column as 1 to 8 ; and that
difference of the arcs in the case of the 4th column to the difference in the

2 16

case of the 1st column as ^ to 7, or as 86 J to 4280 ; put in these

cases 1 and 8 for the velocities, and 85 1 and 4280 for the differences of
the arcs, and A + C will be — S5|, and 8A -f 640 ==• 4280 or A + SC
= 535 ; and then by reducing these equations, there will come out TC =
449^ and C = 64T\ and A = 21f ; and therefore the resistance, which is
as TVAV + fCV2, will become as 13T6TV + 48/^Y2. Therefore in the
case of the 4th column, where the velocity was 1, the whole resistance is to
its part proportional to the square of the velocity as 13T6T + 48/F or
61 }f to 48/e ; and therefore the resistance of the pendulum in water is to
that part of the resistance in air, which is proportional to the square of the
velocity, and which in swift motions is the only part that deserves consid
eration, as 61}^ to 4S/g and 535 to 1} conjunctly, that is, as 571 to 1.
If the whole thread of the pendulum oscillating in the water had been im
mersed, its resistance would have been still greater ; so that the resistance
of the pendulum oscillating in the water, that is, that part which is pro
portional to the square of the velocity, and which only needs to be consid
ered in swift bodies, is to the resistance of the same whole pendulum, oscil
lating in air with the same velocity, as about 850 to 1, that is as, the den
sity of water to the density of air, nearly.

In this calculation we ought also to have taken in that part of the re
sistance of the pendulum in the water which was as the square of the ve
locity ; but I found (which will perhaps seem strange) that the resistance
in the water was augmented in more than a duplicate ratio of the velocity.
In searching after the cause, I thought upon this, that the vessel was toe

320 THE MATHEMATICAL PRINCIPLES [BOOK II.

narrow for the magnitude of the pendulous globe, and by its narrowness
obstructed the motion of the water as it yielded to the oscillating globe.
For when I immersed a pendulous globe, whose diameter was one inch only,
the resistance was augmented nearly in a duplicate ratio of the velocity,
I tried this by making a pendulum of two globes, of which the lesser and
lower oscillated in the water, and the greater and higher was fastened to
the thread just above the water, and, by oscillating in the air, assisted the
motion of the pendulum, and continued it longer. The experiments made
by this contrivance proved according to the following table.
Arc descr. in first descent . .16.8. 4. 2.1.1.1
Arc descr. in last ascent . . 12 . 6 . 3 . li . J . | . T3F
Dif. of arcs, proport. to 1 . pi i

motion lost $ T • «r • T*

Number of oscillations... 3f . 6j . 12^. 211 . 34 . 53 . 62)

In comparing the resistances of the mediums with each other, I also
caused iron pendulums to oscillate in quicksilver. The length of the iron
wire was about 3 feet, and the diameter of the pendulous globe about i of
an inch. To the wire, just above the quicksilver, there was fixed another
leaden globe of a bigness sufficient to continue the motion of the pendulum
for some time. Then a vessel, that would hold about 3 pounds of quick
silver, was filled by turns with quicksilver and common water, that, by
making the pendulum oscillate successively in these two different fluids, I
might find the proportion of their resistances ; and the resistance of the
quicksilver proved to be to the resistance of water as about 13 or 14 to 1 ;
that is. as the density of quicksilver to the density of water. When I made
use of a pendulous globe something bigger, as of one whose diameter was
about ^ or | of an inch, the resistance of the quicksilver proved to be to
the resistance of the water as about 12 or 10 to 1. But the former experi
ment is more to be relied on, because in the latter the vessel was too nar
row in proportion to the magnitude of the immersed globe; for the vessel
ought to have been enlarged together with the globe. I intended to have
repeated these experiments with larger vessels, and in melted metals, and
other liquors both cold and hot ; but I had not leisure to try all: and be
sides, from what is already described, it appears sufficiently that the resist
ance of bodies moving swiftly is nearly proportional to the densities of
the fluids in which they move. I do not say accurately ; for more tena
cious fluids, of equal density, will undoubtedly resist more than those that
are more liquid ; as cold oil more than warm, warm oil more than rain
water, and water more than spirit of wine. But in liquors, which are sen
sibly fluid enough, as in air, in salt and fresh water, in spirit of wine, of
turpentine, and salts, in oil cleared of its fseces by distillation and warmed,
in oil of vitriol, and in mercury, and melted metals, and any other such
like, that are fluid enough to retaia for some time the motion impressed

SEC. VI.J OF NATURAL PHILOSOPHY. 321

upon them by the agitation of the vessel, and which being poured out are
easily resolved into drops, I doubt not but the rule already laid down may
be accurate enough, especially if the experiments be made with larger
pendulous bodies and more swiftly moved.

Lastly, since it is the opinion of some that there is a certain ^ethereal
medium extremely rare and subtile, which freely pervades the pores of all
bodies ; and from such a medium, so pervading the pores of bodies, some re
sistance must needs arise; in order to try whether the resistance, which wre
experience in bodies in motion, be made upon their outward superficies only,
or whether their internal parts meet with any considerable resistance upon
their superficies, I thought of the following experiment I suspended a
round deal box by a thread 11 feet long, on a steel hook, by means of a ring
of the s-ime metal, so as to make a pendulum of the aforesaid length. The
hook had a sharp hollowr edge on its upper part, so that the upper arc of
the ring pressing on the edge might move the more freely ; and the thread
was fastened to the lower arc of the ring. The pendulum being thus pre
pared, I drew it aside from the perpendicular to the distance of about 6
feet, and that in a plane perpendicular to the edge of the hook, lest the
ring, while the pendulum oscillated, should slide to and fro on the edge of
the hook : for the point of suspension, in which the ring touches the hook,
ought to remain immovable. I therefore accurately noted the place to
which the pendulum was brought, and letting it go, I marked three other
places, to which it returned at the end of the 1st, 2d, and 3d oscillation.
This I often repeated, that I might find those places as accurately as pos
sible. Then I filled the box with lead and other heavy metals that were
near at hand. But, first, I weighed the box when empty, and that pnrt of
the thread that went round it, and half the remaining part, extended be
tween the hook and the suspended box ; for the thread so extended always
acts upon the pendulum, when drawn aside from the perpendicular, with half
its weight. To this weight I added the weight of the air contained in the
box And this whole weight was about -fj of the weight of the box when
filled writh the metals. Then because the box when full of the metals, by ex
tending the thread with its weight, increased the length of the pendulum,
f shortened the thread so as to make the length of the pendulum, when os
cillating, the same as before. Then drawing aside the pendulum to the
place first marked, and letting it go, I reckoned about 77 oscillations before
the box returned to the second mark, and as many afterwards before it came
to the third mark, and as many after that before it came to the fourth
xnark. From whence I conclude that the whole resistance of the box, when
full, had not a greater proportion to the resistance of the box, when empty,
than 78 to 77. For if their resistances were equal, the box, when full, by
reason of its vis insita, which was 78 times greater than the vis tfuritoof
the same when empty, ought to have continued its oscillating motion so

21

322 THE MATHEMATICAL PRINCIPLES | BOOK II.

much the longer, and therefore to have returned to those marks at the end
of 78 oscillations. But it returned to them at the end of 77 oscillations.

Let, therefore, A represent the resistance of the box upon its external
superficies, and B the resistance of the empty box on its internal superficies ;
and if the resistances to the internal parts of bodies equally swift be as the
matter, or the number of particles that are resisted, then 78B will be the
resistance made to the internal parts of the box, when full ; and therefore
the whole resistance A + B of the empty box will be to the whole resist
ance A + 7SB of the full box as 77 to 78, and, by division, A + B to 77B
as 77 to 1 ; and thence A + B to B as 77 X 77 to 1, and, by division
again, A to B as 5928 to 1. Therefore the resistance of the empty box in
its internal parts will be above 5000 times less than the resistance on its
external superficies. This reasoning depends upon the supposition that the
greater resistance of the full box arises not from any other latent cause,
but only from the action of some subtile fluid upon the included metal.

This experiment is related by memory, the paper being lost in which I
had described it ; so that I have been obliged to omit some fractional parts,
which are slipt out of my memory ; and I have no leisure to try it again.
The first time I made it, the hook being weak, the full box was retarded
sooner. The cause I found to be, that the hook was not strong enough to
bear the weight of the box : so that, as it oscillated to and fro, the hook
was bent sometimes this and sometimes that way. I therefore procured a
hook of sufficient strength, so that the point of suspension might remain
unmoved, and then all things happened as is above described.

SEC. VI I.] OF NATURAL PHILOSOPHY. 323

SECTION VII.

Of the, motion of fluids, and the resistance made to projected bodies.

PROPOSITION XXXII. THEOREM XXVI.

Suppose two similar systems of bodies consisting of an equal number of
particles, and let the correspondent particles be similar and propor
tional, each in, one system to each in the other, and have a like situa
tion among themselves, and the same given ratio of density to each
other ; and let them begin to move anwng themselves in proportional
times, and with like motions (that is, those in one system among one
another, and those in the other among one another). And if the par
ticles that are in the same system do not touch otte another, except ir
the moments of reflexion ; nor attract, nor repel each other, except with
accelerativeforc.es that are as the diameters of the correspondent parti
cles inversely, and the squares of the velocities directly ; I say, that the
particles of those systems will continue to move among themselves wit It
like motions and in proportional times.

Like bodies in like situations are said to be moved among themselves
with like motions and in proportional times, when their situations at the
end of those times are always found alike in respect of each other ; as sup
pose we compare the particles in one system with the correspondent parti
cles in the other. Hence the times will be proportional, in which similar
and proportional parts of similar figures will be described by correspondent
particles. Therefore if we suppose two systems of this kind; the corre
spondent particles, by reason of the similitude of the motions at their
beginning, will continue to be moved with like motions, so long as they
move without meeting one another ; for if they are acted on by no forces,
they will go on uniformly in right lines, by the 1st Law. But if they do
agitate one another with some certain forces, and those forces are as the
diameters of the correspondent particles inversely and the squares of the
velocities directly, then, because the particles are in like situations, and
their forces are proportional, the whole forces with which correspondent
particles are agitated, and which are compounded of each of the agitating
forces (by Corol. 2 of the Laws), will have like directions, and have the
same effect as if they respected centres placed alike among the particles ;
and those whole forces will be to each other as the several forces which
compose them, that is, as the diameters of the correspondent particles in
versely, and the squares of the velocities directly : and therefore will cans**

3^4 THE MATHEMATICAL PRINCIPLES [BOOK 11.

correspondent particles to continue to describe like figures. These things
will be so (by Cor. 1 and S, Prop. IV.; Book 1), if those centres are at rest
but if they are moved, yet by reason of the similitude of the translations,
their situations among the particles of the system will remain similar , so
that the changes introduced into the figures described by the particles will
still be similar. So that the motions of correspondent and similar par
ticles will continue similar till their first meeting with each other ; and
thence will arise similar collisions, and similar reflexions; which will again
beget similar motions of the particles among themselves (by what was just
now shown), till they mutually fall upon one another again, and so on ad
infinitum.

COR. 1. Hence if any two bodies, which are similar and in like situations
to the correspondent particles of the systems, begin to move amongst them
in like manner and in proportional times, and their magnitudes and densi
ties be to each other as the magnitudes and densities of the corresponding
particles, these bodies will continue to be moved in like manner and in
proportional times: for the case of the greater parts of both systems and of
the particles is the very same.

COR. 2. And if all the similar and similarly situated parts of both sys
tems be at rest among themselves ; and two of them, which are greater than
the rest, and mutually correspondent in both systems, begin to move in
lines alike posited, with any similar motion whatsoever, they will excite
similar motions in the rest of the parts of the systems, and will continue
to move among those parts in like manner and in proportional times ; and
will therefore describe spaces proportional to their diameters.

PROPOSITION XXXIII. THEOREM XXVII.

The same things biting supposed, I say, that the greater parts of the
systems are resisted in a ratio compounded of the duplicate ratio of
their velocities, and the duplicate ratio of their diameters, and Ihe sim
ple ratio of the density of the parts of the systems.
For the resistance arises partly from the centripetal or centrifugal, forces
with which the particles of the system mutually act on each other, partly
from the collisions and reflexions of the particles and the greater parts.
The resistances of the first kind are to each other as the whole motive
forces from which they arise, that is, as the whole accelerative forces and
the quantities of matter in corresponding parts ; that is (by the sup
position), as the squares of the velocities directly, and the distances of the
corresponding particles inversely, and the quantities of matter in the cor
respondent parts directly : and therefore since the distances of the parti
cles in one system are to the correspondent distances of the particles of the
;ther S3 the diameter of one particle or part in *he former system to the

SEC. VII.] OF NATURAL PHILOSOPHY. C>2"

diameter of the correspondent particle or part in the other, and since the
quantities of matter are as the densities of the parts and the cubes of the
diameters ; the resistances arc to each other as the squares of the velocities
and the squares of the diameters and the densities of the parts of the sys
tems. Q.E.D. The resistances of the latter sort are as the number of
sorrespondent reflexions and the forces of those reflexions conjunctly ; but
the number of the reflexions are to each other as the velocities of the cor
responding parts directly and the spaces between their reflexions inversely.
And the forces of the reflexions are as the velocities and the magnitudes
and the densities of the corresponding parts conjunctly ; that is, as the ve
locities and the cubes of the diameters and the densities of the parts. And,
joining all these ratios, the resistances of the corresponding parts are to
each other as the squares of the velocities and the squares of the diameters
and the densities of the parts conjunctly. Q.E.T).

COR. 1. Therefore if those systems are two elastic fluids, like our air,
and their parts are at rest among themselves ; and two similar bodies pro
portional in magnitude and density to the parts of the fluids, and similarly
gituated among those parts, be any how projected in the direction of lines
similarly posited ; and the accelerative forces with which the particles of
the fluids mutually act upon each other are as the diameters of the bodies
projected inversely and the squares of their velocities directly ; those bodies
will excite similar motions in the fluids in proportional times, and will de
scribe similar spaces and proportional to their diameters.

COR. 2. Therefore in the same fluid a projected body that moves swiftly
meets with a resistance that is, in the duplicate ratio of its velocity, nearly.
For if the forces with which distant particles act mutually upon one
another should be augmented in the duplicate ratio of the velocity, the
projected body would be resisted in the same duplicate ratio accurately ;
and therefore in a medium, whose parts when at a distance do not act mu
tually with any force on one another, the resistance is in the duplicate ra
tio of the velocity accurately. Let there be, therefore, three mediums A,
B, C, consisting of similar and equal parts regularly disposed at equal
distances. Let the parts of the mediums A and B recede from each other
with forces that are among themselves as T and V ; and let the parts of
the medium C be entirely destitute of any such forces. And if four equal
bodies D, E, P7 G, move in these mediums, the two first D and E in the
two first A and B, and the other two P and G in the third C ; and if the
velocity of the body D be to the velocity of the body E, and the velocity
of the body P to the velocity of the body G, in the subduplicate ratio of
the force T to the force V ; the resistance of the body D to the resistance
of the body E, and the resistance of the body P to the resistance of the
body G, will be in the duplicate ratio of the velocities ; and therefore the
resistance of the body D will be to the resistance of the body P as the re-

326 THE MATHEMATICAL PRINCIPLES [BOOK II

sistance of the body E to the resistance of the body G. Let the bodies 1)
and F be equally swift, as also the bodies E and G ; and, augmenting the
velocities of the^bodies D arid F in any ratio, and diminishing the forces
of the particles of the medium B in the duplicate of the same ratio, the
medium B will approach to the form and condition of the medium C at
pleasure ; and therefore the resistances of the equal and equally swift
bodies E and G in these mediums will perpetually approach to equality
so that their difference will at last become less than any given. There
fore since the resistances of the bodies D and F are to each other as the
resistances of the bodies E and G, those will also in like manner approach
to the ratio of equality. Therefore the bodies 1) and F, when they move
with very great swiftness, meet with resistances very nearly equal; and
therefore since the resistance of the body F is in a duplicate ratio of the
velocity, the resistance of the body D will be nearly in the same ratio.

Con. 3. The resistance of a body moving very swift in an elastic fluid
is almost the same as if the parts of the fluid were destitute of their cen
trifugal forces, and did not fly from each other; if so be that the elasti
city of the fluid arise from the centrifugal forces of the particles, and the
velocity be so great as not to allow the particles time enough to act.

COR. 4. Therefore, since the resistances of similar and equally swift
bodies, in a medium whose distant parts do not fly from each other, are as
the squares of the diameters, the resistances made to bodies moving with
very great and equal velocities in an elastic fluid will be as the squares of
the diameters, nearly.

COR. 5. And since similar, equal, and equally swift bodies, moving
through mediums of the same density, whose particles do not fly from each
other mutually, will strike against an equal quantity of matter in equal
times, whether the particles of which the medium consists be more and
smaller, or fewer and greater, and therefore impress on that matter an equal
quantity of motion, and in return (by the 3d Law of Motion) suffer an
equal re-action from the same, that is, are equally resisted ; it is manifest,
also, that in elastic fluids of the same density, when the bodies move with
extreme swiftness, their resistances are nearly equal, whether the fluids
consist of gross parts, or of parts ever so subtile. For the resistance of
projectiles moving with exceedingly great celerities is not much diminished
by the subtilty of the medium.

COR. G. All these things are so in fluids whose elastic force takes its rise
from the centrifugal forces of the particles. But if that force arise from
some other cause, as from the expansion of the particles after the manner
of wool, or the boughs of trees, or any other cause, by which the particles
are hindered from moving freely among themselves, the resistance, by
reason of the lesser fluidity of the medium, will be greater than in the
Corollaries above.

SEC. VII.

OF NATURAL PHILOSOPHY.

32?

K

L, P

O

PROPOSITION XXXIV. THEOREM XXV1I1.

If iu a rare medium, consisting of equal particles freely disposed at
equal distances from each other, a globe and a cylinder described on
equal diameters move with equal velocities in the. direction of the axis
of the cylinder, the resistance of the globe ivill be but half so great an
that of the cylinder.
For since the action of the medi
um upon the body is the same (by
Cor. 5 of the Laws) whether the body
move in a quiescent medium, or
whether the particles of the medium
impinge with the same velocity upon
the quiescent body, let us consider
the body as if it were quiescent, and
see with what force it would be im-
pelled by the moving medium. Let, therefore, ABKI represent a spherical
body described from the centre C with the semi-diameter CA, and let the
particles of the medium impinge with a given velocity upon that spherical
body in the directions of right lines parallel to AC : and let FB be one of
those right lines. In FB take LB equal to the semi-diameter CB, and
draw BI) touching the sphere in B. Upon KG and BD let fall the per
pendiculars BE, LD ; and the force with which a particle of the medium,
impinging on the globe obliquely in the direction FB, would strike the
globe in B, will be to the force with which the same particle, meeting the
cylinder ONGQ, described about the globe with the axis ACI, would strike
it perpendicularly in b, as LD to LB, or BE to BC. Again ; the efficacy
of this force to move the globe, according to the direction of its incidence
FB or AC, is to the efficacy of the same to move the globe, according to
the direction of its determination, that is, in the direction of the right line
BC in which it impels the globe directly, as BE to BC. And, joining
these ratios, the efficacy of a particle, falling upon the globe obliquely in
the direction of the right line FBy to move the globe in the direction of its
incidence, is to the efficacy of the same particle falling in the same line
perpendicularly on the cylinder, to move it in the same direction, as BE2
to BC3. Therefore if in 6E, which is perpendicular to the circular base of
the cylinder NAO, and equal to the radius AC, we take £H equal to

BEa

-— • ; then 6H will be to 6E as the effect of the particle upon the globe t<?
\~i\j

the effect of the particle upon the cylinder. Arid therefore the solid which
is formed by all the right lines 6H will be to the solid formed by all the
right lines />E as the effect of all the particles upon the globe to the effect
of all the particles upon the cylinder. But the former of these solids is a

328

THE MATHEiAlATICAL PRINCIPLES

[BooK li.

paraboloid whose vertex is C, its axis CA, and latus rectum CA, and the
latter solid is a cylinder circumscribing the paraboloid ; and it is knowr
that a paraboloid is half its circumscribed cylinder. Therefore the whole
force of the medium upon the globe is half of the entire force of the same
upon the cylinder. And therefore if the particles of the medium are at
rest, and the cylinder and globe move with equal velocities, the resistance
of the globe will be half the resistance of the cylinder. Q.E.D.

SCHOLIUM.

By the same method other figures may be compared together as to their
resistance; and those may be found which are most apt to continue their
motions in resisting mediums. As if upon the circular base CEBH from
the centre O, with thy radius OC, and the altitude OD, one would construct
a frustum CBGF of a cone, which should meet with less resistance than
any other frustum constructed with the same base and altitude, and going
forwards towards D in the direction of its axis : bisect the altitude OD in
U,, and produce OQ, to S, so that QS may be equal to Q,C, and S will be
the vertex of the cone whose frustum is sought.
r

J

Whence, by the bye, since the angle CSB is always acute, it follows, that,
if the solid ADBE be generated by the convolution of an elliptical or oval
figure ADBE about its axis AB, and the generating figure be touched by-
three right lines FG, GH, HI, in the points F, B, and I, so that GH shall
be perpendicular to the axis in the point of contact B, arid FG, HI may be
inclined to GH in the angles FGB, BHI of 135 degrees: the solid arising
from the convolution of the figure ADFGH1E about the same axis AB
will be less resisted than the former solid; if so be that both move forward
in the direction of their axis AB, and that the extremity B of each go
foremost. Which Proposition I conceive may be of use in the building of
ships.

If the figure DNFG be such a curve, that if, from any point thereof, as
N, the perpendicular NM be let fall on the axis AB, and from the given
point G there be drawn the right line GR parallel to a right line touching
the figure in N, and cutting the axis produced in R, MN becomes to GR
as GR,3 to 4BR X GB2, the solid described, by the revolution of this figure

SEC. Vll.J OF NATURAL PHILOSOPHY. 32S

about its axis AB, moving in the before-mentioned rare medium from A
towards B, will be less resisted than any other circular solid whatsoever,
described of the same length and breadth.

PROPOSITION XXXV. PROBLEM VII.

If a rare medium consist of very small quiescent particles of equal mag
nitudes, and freely disposed at equal distances from one another : to
jind the resistance of a globe moving uniformly forward in this
medium.

CASE 1. Let a cylinder described with the same diameter and altitude be
conceived to go forward with the same velocity in the direction of its axis
through the same medium ; and let us suppose that the particles of the
medium, on which the globe or cylinder falls, fly back with as great a force
of reflexion as possible. Then since the resistance of the globe (by the last
Proposition) is but half the resistance of the cylinder, and since the globe
is to the cylinder as 2 to 3, and since the cylinder by falling perpendicu
larly on the particles, and reflecting them with the utmost force, commu
nicates to them a velocity double to its own; it follows that the cylinder.
in moving forward uniformly half the length of its axis, will communicate
a motion to the particles which is to the whole motion of the cylinder as
the density of the medium to the density of the cylinder ; and that the
globe, in the time it describes one length of its diameter in moving uni
formly forward, will communicate the same motion to the particles ; and
in the time that it describes twro thirds of its diameter, will communicate
a motion to the particles which is to the whole motion of the globe as the
density of the medium to the density of the globe. Arid therefore the
globe meets with a resistance, which is to the force by which its whole mo
tion may be either taken away or generated in the time in which it de
scribes two thirds of its diameter moving uniformly forward, as the den
sity of the medium to the density of the globe.

CASE 2. Let us suppose that the particles of the medium incident on
the globe or cylinder are not reflected ; and then the cylinder falling per
pendicularly on the particles will communicate its own simple velocity to
them, and therefore meets a resistance but half so great as in the former
case, and the globe also meets with a resistance but half so great.

CASE 3. Let us suppose the particles of the medium to fly back from
the globe with a force which is neither the greatest, nor yet none at all, but
with a certain mean force ; then the resistance of the globe will be in the
same mean ratio between the resistance in the first case and the resistance
in the second. Q.E.I.

COR. 1. Hence if the globe and the particles are infinitely hard, and
destitute of all elastic force, and therefore of all force of reflexion ; thf
resistance of the globe will be to the force by which its whole motion may

330 THE MATHEMATICAL PRINCIPLES [BOOK I)

be destroyed or generated, in the time that the globe describes four third
parts of its diameter, as the density of the medium to the density of the
^lobe.

Con. 2. The resistance of the globe, cceteris paribus, is in the duplicate
ratio of the velocity.

CUR. 3. The resistance of the globe, cocteris paribus, is in the duplicate
ratio of the diameter.

COR. 4. The resistance of the globe is, cceteris paribus, as the density of
the medium.

COR, 5. The resistance of the globe is in a ratio compounded of the du
plicate ratio of the velocity, arid the duplicate ratio of the diameter, and
the ratio of the density of the medium.

COR. 6. The motion of the globe and its re
sistance may be thus expounded Let AB be the
time in which the globe may, by its resistance
uniformly continued, lose its whole motion.
Erect AD, BC perpendicular to AB. J ,et BC be
that whole motion, and through the point C, the
asymptotes being AD, AB, describe the hyperbola
CF. Produce AB to any point E. Erect the perpendicular EF meeting
the hyperbola in F. Complete the parallelogram CBEG, and draw AF
meeting BC in H. Then if the globe in any time BE, with its first mo
tion BC uniformly continued, describes in a non-resisting medium the space
CBEG expounded by the area of the parallelogram, the same in a resisting
medium will describe the space CBEF expounded by the area of the hv-
perbola; and its motion at the end of that time will be expounded by EF,
the ordinate of the hyperbola, there being lost of its motion the part FG.
And its resistance at the end of the same time will be expounded by the
length BH, there being lost of its resistance the part CH. All these things
appear by Cor. 1 and 3, Prop. V., Book II.

COR. 7. Hence if the globe in the time T by the resistance R uniformly
continued lose its whole motion M, the same globe in the time t in a
resisting medium, wherein the resistance R decreases in a duplicate

/M

ratio of the velocity, will lose out of its motion M the part ,.i ' the

TM

part rn . ; remaining ; and will describe a space which is to the space de
scribed in the same time t, with the uniform motion M, as the logarithm of

T + t
the number — ^.— multiplied by the number 2,302585092994 is to the

number ^ because the hyperbolic area BCFE is to the rectangle BCGE
in that proportion.

SEC. VII.]

OF NATURAL PHILOSOPHY.

331

SCHOLIUM.

I have exhibited in this Proposition the resistance and retardation of
spherical projectiles in mediums that are not continued, and shewn that
this resistance is to the force by which the whole motion of the globe may be
destroyed or produced in the time in which the globe can describe two thirds
of its diameter, with a velocity uniformly continued, as the density of the
medium to the density of the globe, if so be the globe and the particles of
the medium be perfectly elastic, and are endued with the utmost force of
reflexion ; and that this force, where the globe and particles of the medium
are infinitely hard and void of any reflecting force, is diminished one half.
But in continued mediums, as water, hot oil, and quicksilver, the globe as
it passes through them does not immediately strike against all the parti
cles of the fluid that generate the resistance made to it, but presses only
the particles that lie next to it, which press the particles beyond, which
press other particles, and so on ; and in these mediums the resistance is di
minished one other half. A globe in these extremely fluid mediums meets
with a resistance that is to the force by which its whole motion may be
destroyed or generated in the time wherein it can describe, with that mo
tion uniformly continued, eight third parts of its diameter, as the density
of the medium to the density of the globe. This I shall endeavour to shew
in what follows.

PROPOSITION XXXVI. PROBLEM VIII.

To define the motion of water running out of a cylindrical vessel through

a hole made at the bottom.

Let AC D B be a cylindrical vessel, AB the mouth p = Q:

of it, CD the bottom p irallel to the horizon, EF a
circular hole in the middle of the bottom, G the
c-?ritre of the hole, and GH the axis of the cylin- Kj
cler perpendicular to the horizon. And suppose a
cylinder of ice APQ,B to be of the same breadth
with the cavity of the vessel, and to have the same
axis, and to descend perpetually with an uniform
motion, and that its parts, as soon as they touch the
superficies AB, dissolve into water, and flow
( wn by their weight into the vessel, and in their
fall compose the cataract or column of water
ABNFEM, passing through the hole EF, and filling up the same exactly.
Let the uniform velocity of the descending ice and of the contiguous water
in the circle AB be that which the water would acquire by falling through
the space IH ; and let IH and HG lie in the same right line ; and through

332 THE MATHEMATICAL PRINCIPLES [BOOK Jl

the point I let there be drawn the right line KL parallel to the horizon
and meeting the ice on both the sides thereof in K and L. Then the ve
locity of the water running out at the hole EF will be the same that it
would acquire by falling from I through the space IG. Therefore, by
Galih'cJ's Theorems, IG will be to IH in the duplicate ratio of the velo
city of the water that runs out at the hole to the velocity of the wrater in
the circle AB, that is, in the duplicate ratio of the circle AB to the circle
EF ; those circles being reciprocally as the velocities of the water which
in the same time and in equal quantities passes severally through each of
them, and completely fills them both. We are now considering the velo
city with which the water tends to the plane of the horizon. But the mo
tion parallel to the same, by which the parts of the falling water approach to
each other, is not here taken notice of; since it is neither produced by
gravity, nor at all changes the motion perpendicular to the horizon which the
gravity produces. We suppose, indeed, that the parts of the water cohere
a little, that by their cohesion they may in falling approach to each othei
with motions parallel to the horizon in order to form one single cataract,
and to prevent their being divided into several : but the motion parallel to
the horizon arising from this cohesion does not come under our present
consideration.

CASE 1. Conceive now the w^hole cavity in the vessel, wrhich encompasses
the falling water ABNFEM, to be full of ice, so that the water may pass
through the ice as through a funnel. Then if the water pass very near to
the ice only, without touching it; or, which is the same tiling, if by rea
son of the perfect smoothness of the surface of the ice, the water, though
touching it. glides over it writh the utmost freedom, and without the le-ast
resistance; the water will run through the hole EF with the same velocity
as before, and the whole weight of the column of water ABNFEM will be
all taken up as before in forcing out the water, and the bottom of the vessel
will sustain the weight of the ice encompassing that column.

Let now the ice in the vessel dissolve into water ; yet will the efflux of
the water remain, as to its velocity, the same as before. It will not be
less, because the ice now dissolved will endeavour to descend ; it will not
be greater, because the ice. now become water, cannot descend without hin
dering the descent of other water equal to its own descent. The same force
ought always to generate the same velocity in the effluent water.

But the hole at the bottom of the vessel, by reason of the oblique mo
tions of the particles of the effluent water, must be a little greater than before,
For now the particles of the water do not all of them pass through the
hole perpendicularly, but, flowing down on all parts from the sides of the
vessel, and converging towards the hole, pass through it with oblique mo
tions : r,r,d in tending downwards meet in a stream whose diameter is a little
smaller below the hole than at the hole itself : its diameter being to the

SEC. V1L!

OF NATURAL PHILOSOPHY.

333

diameter of the hole as 5 to 6, or as 5^ to 6|, very nearly, if I took the
measures of those diameters right. I procured a very thin flat plate, hav
ing a hole pierced in the middle, the diameter of the circular hole being
f parts of an inch. And that the stream of running waters might not be
accelerated in falling, and by that acceleration become narrower, I fixed
this plate not to the bottom, but to the side of the vessel, so us to make the
water go out in the direction of a line parallel to the horizon. Then, when
the vessel was full of water, I opened the hole to let it run out ; and the
diameter of the stream, measured with great accuracy at the distance of
about half an inch from the hole, was f J- of an inch. Therefore the di
ameter of this circular hole was to the diameter of the stream very nearly
as 25 to 21. So that the water in passing through the hole converges on
all sides, and, after it has run out of the vessel, becomes smaller by converg
ing in that manner, and by becoming smaller is accelerated till it comes to
the distance of half an inch from the hole, and at that distance flows in a
smaller stream and with greater celerity than in the hole itself, and this
in the ratio of 25 X 25 to 21 X 21, or 17 to 12, very nearly ; that is, in
about the subdaplicate ratio of 2 to 1. Now it is certain from experiments,
that the quantity of water running out in a given time through a circular
hole made in the bottom of a vessel is equal to the quantity, which, flow
ing with the aforesaid velocity, would run out in the same time through
another circular hole, whose diameter is to the diameter of the former as
21 to 25. And therefore that running water in passing through the
hole itself has a velocity downwards equal to that which a heavy body
would acquire in falling through half the height of the stagnant water in
the vessel, nearly. But, then, after it has run out, it is still accelerated by
converging, till it arrives at a distance from the hole that is nearly equal to
its diameter, and acquires a velocity greater than the other in about the
subduplicate ratio of 2 to 1 ; which velocity a heavy body would nearly
acquire by falling through the whole height of the stagnant water in the
vessel.

Therefore in what follows let the diameter of
the stream be represented by that lesser hole which
we called EF. And imagine another plane VW
above the hole EF, and parallel to the plane there
of, to be placed at a distance equal to the diame
ter of the same hole, and to be pierced through
with a greater hole ST, of such a magnitude that
a stream which will exactly fill the lower hole EF
may pass through it ; the diameter of which hole
will therefore be to the diameter of the lower hole as 25 to 21, nearly. By
this means the water will run perpendicularly out at the lower hole ; and
the quantity of the water running out will be, according to the magnitude

334 THE MATHEMATICAL PRINCIPLES [BOOK 11

of this last hole, the same, very nearly, which the solution of the Problem
requires. The space included between the two planes and the falling stream
may be considered as the bottom of the vessel. But, to make the solution
more simple and mathematical, it is better to take the lower plane alone
for the bottom of the vessel, and to suppose that the water which flowed
through the ice as through a funnel, and ran out of the vessel through the
hole EF made in the lower plane, preserves its motion continually, and that
the ice continues at rest. Therefore in what follows let ST be the diame
ter of a circular hole described from